This document provides a list of useful links for technology in the English language classroom. It includes links to websites for webquests, interactive whiteboard resources, video resources, working with news articles, English language teaching materials, tools for teachers and students, curriculum resources, and examples of Web 2.0 technologies like course management systems, photo sharing, social networking, and bookmarking sites. The links are organized into categories but not described in detail.
Stephanie Herrera has created a portfolio to showcase her work experience and skills. Her portfolio likely contains examples of work she has completed, as well as her resume and references. The purpose of her portfolio is to help her land new job opportunities in her field by demonstrating her qualifications and abilities to potential employers.
Be healthy through moderate exercise and enough rest. Spend time with family and friends, doing activities you enjoy like sports or comedy. Maintain a positive outlook by reflecting on happy memories, forgiving others, and focusing on the present moment instead of worries. Schedule work and leisure appropriately to maintain a work-life balance.
This document provides a list of useful links for technology in the English language classroom. It includes links to websites for webquests, interactive whiteboard resources, video resources, working with news articles, English language teaching materials, tools for teachers and students, curriculum resources, and examples of Web 2.0 technologies like course management systems, photo sharing, social networking, and bookmarking sites. The links are organized into categories but not described in detail.
Stephanie Herrera has created a portfolio to showcase her work experience and skills. Her portfolio likely contains examples of work she has completed, as well as her resume and references. The purpose of her portfolio is to help her land new job opportunities in her field by demonstrating her qualifications and abilities to potential employers.
Be healthy through moderate exercise and enough rest. Spend time with family and friends, doing activities you enjoy like sports or comedy. Maintain a positive outlook by reflecting on happy memories, forgiving others, and focusing on the present moment instead of worries. Schedule work and leisure appropriately to maintain a work-life balance.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
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- La variación de posición de la recta
- El ángulo que la recta forma con el eje x
30. f (xA + x)
B(x ,y )
B B
y= f ( x)
y
f (xA)
A(x ,y )
A A
xA xA + x
x X
31. f ( xA+ Dx)
B(xB,yB)
f ( xA+ Dx)
B(xB,yB)
y= f ( x)
f (xA)
A(x ,y )
A A
xA+ Dx xA+ Dx
xA
X
32. y f ( xA x) f ( x A ) f ( xA x) f ( xA )
tg
x xA x xA x
f (xA+ x)
y= f ( x)
y
y
f (xA)
A(x ,y )
A A
xA xA+ x
x X
x
33. ¡¡ FIJATE !!
Cuando cambiamos el punto B:
¿Quien se esta haciendo
menor?
¿Quien esta
variando?
¡VOLVAMOS A LA IMAGEN!
34. y f ( xA x) f ( x A ) f ( xA x) f ( xA )
tg
x xA x xA x
B 1
f (xA+ x) B
y= f ( x)
y
y
f (xA)
A(x ,y )
A A
xA xA+ x
x X
x
35. Se hacen menores los
incrementos de las
coordenadas, Dx e Dy
Varia el ángulo que forma la recta
con el eje x; en este caso
aumentando
PROSIGAMOS
37. B y= f ( x)
Dx1>Dx2>Dx
1
B 2
Dy1>Dy2>Dy f (xA+ x) B
y1
2
y
y
Se ve que:
x 0 f (xA)
(Dx tiende a cero)
A(x ,y )
A A
y 0
(Dy tiende a cero)
xA xA+ x
x
x
2
X
x1
Los ángulos a<m<b,son los formados por la recta y el eje Ox (parte +)
38. Cuando el punto B, de la función y=f(x), se
acerca a A:
0
1. x 0 y 0
2º.- El ángulo tomado es el formado
por la recta y la parte positiva del eje
Ox. En este caso:
tg a< tg m < tg b
39. y f ( xA x) f ( x A ) f ( xA x) f ( xA )
tg
x xA x xA x
Y B1 y= f ( x)
B 2
y1
y2
f (x + x) B
A
y
f (x )
A
A(x ,y ) A A
x A x+ xA
x
x 2
X
x
1
40. y f ( xA x) f ( x A ) f ( xA x) f ( xA )
tg
x xA x xA x
f (xA+ x) B
y
f (xA)
A(x ,y )A A
xA xA+ x
x
42. Cuando el punto B se confunde con A:
y f ( xA x) f ( x A ) f ( xA x) f ( xA )
tg
x xA x xA x
Como
x 0 y 0
y 0
tg
x 0
Función (tgr) que NO se puede calcular al ser un valor indetermi-
nado, por lo tanto hay que hallar su límite
43. y f ( xA x) f ( xA )
tg lim x 0 lim x 0
x x
tangente geométrica
Y B1 y= f ( x)
Se toma Dx porque B
2
es la variable inde-
pendiente en la
función y= f(x) B 3
f (x + x) B
A
f (x ) A
A
y
x
x x+ x
A A
X
Al tender B a A, la cuerda que une los puntos, se transformará en la tangen-
te geométrica a la función en el punto A, cuyo ángulo, respecto a la zona
positiva del eje Ox, es r , y el valor de la tg r es la DERIVADA en el punto A.
44. 3
f (x + x) B
A
f (x ) A A
y
x
tangente geométrica
x x+ x
A A
y f ( xA x) f ( xA )
tg lim x 0 lim x 0
x x
45. Cuando el punto B tiende a A, la secante que los
une se convierte en tangente geométrica
formando, con la parte positiva del eje Ox, un
ángulo r:
siendo r= arc tg r, calculando su:
dy y f ( xA x) f ( xA )
tg lim x 0 lim x 0
dx x x
Llamamos DERIVADA primera a la
tga, anotándola, también, como y’, por ello:
dy f (x x) f ( x)
y' tg lim x 0
dx x
46. dy y f (x x) f ( x)
y' tg lim x 0 lim x 0
dx x x
Hallar la primera derivada de la función y = 2x – 5, en el punxo x = 3
Primeramente hallaremos f ( x ) = 2x – 5 que para x = 3 y = f(3)=2.3 - 5
=1
Después f ( x + Dx ) = 2 ( x + Dx ) – 5 ; cambiando x por (x + Dx)
dy f (x x) f ( x) [2( x x) 5] [2 x 5]
y' tg lim x 0 lim x 0
dx x x
2x 2 x 5 2x 5 2 x
lim x 0 lim x 0 lim x 0 2 2
x x
El ángulo r que forma la tangente geométrica con la parte positiva del eje Ox, en el
punto x =3, es r = arc tg 2 = 63º26’6’’
La ecuación de la tangente geométrica , y – y1 = m ( x - x1 ) será, teniendo en
cuenta que x1 = 3 ; y1 = 1 y m = tgr = 2; y – 1 = 2 ( x – 3 ) o sea y = 2x - 5
SOLO, en la función de primer grado, y = ax + b, coincide la función (una recta) con la
tangente geométrica
47. dy y f (x x) f ( x)
y' tg lim x 0 lim x 0
dx x x
Hallar la primera derivada de la función y = -3x + 4, en el punto x = 8
Primeramente hallaremos f ( x ) = -3x + 4 que para x =5 y = f(5) = -3.5 + 4 = -
11
Después f ( x + Dx ) = -3 ( x + Dx ) + 4 ; cambiando x por (x + Dx)
dy f (x
x) f ( x) [ 3( x x) 4] [ 3 x 4]
y' tg lim x 0 lim x 0
dx x x
3x 3 x 4 3x 4 3 x
lim x 0 lim x 0 lim x 0 3 3
x x
El ángulo r que forma la tangente geométrica con la parte positiva del eje Ox, en el
punto x = 5, es r = arc tg (-3) = 108º26’6’’
La ecuación de la tangente geométrica , y – y1 = m ( x - x1 ) ; teniendo en
cuenta que x1 = 5 ; y1 = -11 y m = tgr = -3 ; y – (-11) = -3 ( x – 5 ) o sea
y = -3x + 4
SOLO, en la función de primer grado, y = ax + b, coincide la función (una recta) con la
tangente geométrica
48. RECUERDA
El ángulo r que forma la tangente geométrica con la parte positiva del eje Ox, en
el primer problema, es r = arc tg 2 = 63º26’6’’, ángulo menor de 90º, en el segundo
problema, es r = arc tg (-3) = 108º26’6’’, ángulo mayor de 90º
x f(x)=2x-5 y=2x-5 x f(x)=-3x+4
y=-3x+4
0 -5 0 4
1
Y Y -11
3 =5
10
8 º2
6'6
=6
''
3º 2
A(3,1)
6'6
(0,4)
''
X (parte X (parte X (parte X (parte
negativa) (0,-5)
positiva) negativa) positiva)
A(5,-11)
Si la derivada es positiva el ángulo que forma, la tangente geométrica con la parte positiva del
eje Ox, es menor que el ángulo recto, si es negativa forma un ángulo superior al recto
49. SEGUIMOS
Halla la primera derivada de la función y = 2x2 - 3x + 4, en los puntos
a) x en general; b) x = 1; c) x = -1
Sabemos que f ( x ) = 2x2 - 3x + 4 y que f ( x + Dx ) = 2 ( x + Dx ) 2- 3 (x + Dx)
+4
f (x x) f ( x) [2( x x) 2 3( x x) 4] [2 x 2 3 x 4]
y ' lim x 0 lim x 0
x x
2 x 2 4 x x 2( x) 2 3x 3 x 4 2 x 2 3 x 4 4x x 2 x2 3 x
lim x 0 lim x 0
x x
x 4x 2 x 3
lim x 0 lim x 0 4 x 2 x 3 4 x 2.0 3 4 x 3
x
El ángulo r que forma la tangente geométrica con la parte positiva del eje Ox, en
un punto general x es r = arc tg (4x - 3)
La ecuación de la tangente geométrica , y – y1 = m ( x - x1 ) ; con m = tgr = (4x - 3)
Para el punto x 1= 1 y1 = f(1)= 2 . 12 – 3 . 1 + 4 = 3 y’ = m = tg r= 4 . 1 – 3 = 1
Ecuac. de la tangente geométrica y – 3 = 1 . ( x – 1) Y y = x + 2 con =arctg 1 =
45º
Para el punto x 2= -1 y2 = f (-1)= 2 . (-1) 2– 3 . (-1) + 4 = 9 y’ = m = tg r= 4 . (-1) – 3
= -7
Ec. tangente geométrica y – 9 = -7 . ( x – (-1)) y = -7x +2 con r =arctg (-7)=
50. Y
y f ( x) 2 x 2 3x 4
y' = 4.x - 3
(-1,9)
Cuando x = 1 y' = 4.1 - 3 = 1
x f(x) tg = y'=1 = 45°
-1 9 (2,6)
98
0 4 °7
0'75 2'25 '4 Cuando x = -1 y' = 4.(-1) - 3 = -7
(0,4) 8'
1 3 '
tg = y'=-7 = 98°7'48''
2 6 (1,3)
45°
X (parte
y=x+2 positiva)
y = -7x + 2
FIJATE, en el punto (1 , 3) y’ = 1, valor POSITIVO, tg r>0 y r<90o; en el punto (-1,9) y’ =
-7, valor NEGATIVO,tg r<0 y r>90o, ¿habrá algún punto en donde y’= 0?
51. Y
(-1,9) y f ( x) 2 x 2 3x 4
y' = 4x - 3
x f(x)
Vértice,cuando y' = 0
-1 9
0 = 4x - 3
0 4
(2,6) x =0'75 f(x) = y =2'25
0'75 2'25 98
1 3 °7 Es el valor MÍNIMO
'4
2 6 (0,4) 8' que toma la función
'
y=2'25
(1,3)
tg =0 =0°
45°
y-2'25 = 0.(x - 0'75)
y-2'25 = 0
X (parte positiva) y=2'25
y=x+2
y = -7x + 2
RECUERDA, en el valor de x que hace tg r = y’ = 0, tendremos un MÍNIMO si en su entorno, a
la izquierda, las tg r tienen valores negativos, pasando a positivos, a la derecha
52. Detalle del punto en donde se encuentra el MÍNIMO y en donde tgr=0
y f ( x) 2 x 2 3x 4
(1,3) y=2'25
(0'75,2'25)
45°
RECUERDA, en el valor de x que hace tg r = y’ = 0, tendremos un MÍNIMO si en su entorno, a
la izquierda, las tg r tienen valores negativos, pasando a positivos, a la derecha
53. Halla la primera derivada de la función x = - 3t2 + 6t + 4, en los
puntos a) t general; b) t = 2; c) t = -1; d) en donde estará el mínimo
Sabemos que f ( t ) = - 3t2 + 6t + 4 y que f ( t + Dt ) = - 3 ( t + Dt ) 2+ 6 (t + Dt)
+4
f (t t ) f (t ) [ 3(t t ) 2 6(t t ) 4] [ 3t 2 6t 4]
x' lim t 0 lim t 0
t t
3t 2 6t t 6( t ) 2 6t 6 t 4 3t 2 6t 4 6t t 6( t ) 2 6 t
lim t 0 lim t 0
t t
t 6t 6 t 6
lim t 0 lim x 0 6t 6 t 6 6t 6.0 6 6t 6
t
El ángulo r que forma la tangente geométrica con la parte positiva del eje OX, en
un punto general es r = arc tg (- 6t + 6)
La ecuación de la tangente geométrica , y – y1 = m ( t - t1 ) ; con m = tgr = (- 6t + 6)
Para el punto t 1= 2 x1 = (-3) . 22 + 6 . 2 + 4 = 4 x’ = m = tg r= - 6 . 2 + 6 = - 6
Ec. tang. geométrica x – 4 = (- 6) . ( t – 2) Y x = - 6t + 16 con r = arctg(- 6) =
99º27’44’’
Para el punto t 2= - 1 x2 = (-3) . (-1) 2+ 6 . (-1) + 4 = -5 x’ = m = tg r= (- 6) . (-1) + 6
= 12
Ec. Tang. geométrica x + 5 = 12 . ( t – (-1)) Yx = 12t +7 con r =arctg 12 = 85º14’11’’
Habrá un ¿mínimo? cuando x’= 0 Y - 6t + 6 = 0Y t 3= 1 x3 = (-3) . 1 + 6 . 1 + 4 = 7
Ec. tangente geométrica x – 7 = 0. ( t – 1 ) y x = 7 con r = arctg 0 = 0º
54. Fijaros, además, cuando x’>0 la función crece, cuando x’ es negativa decrece
x x' = -6t + 6
x=7 (1,7)
x f(t)
f(x)
Vértice cuando x' = 0
1 7 0 = -6t + 6
2 4 t=1 f(t) = x = 7
x = 12t + 7
(2,4) 3 -5
(0,4)
0 4 Este valor es el MÁXIMO que
99
°2 -1 -5 toma la función
7'
44
'' Cuando t = 2 y' = (-6).2 + 6 = -6 -6
x' = (-6).2 + 6 =
85
85
°1
4
4''
11
1'
tg = x'= -6 = 99°27'44'' > 90°
''
t (parte
x f (t ) 3t 2 6t 4
positiva) Cuando t = -1
Cuando t = -1
x' = (-6).(-1) + 6 = 12
x' = (-6).(-1) + 6 = 12
tg = x'=12 = 85°14'11'' < 90°
tg = x'=12 = 85°14'11'' < 90°
Cuando t = 1 x' = (-6).(1) + 6 = 0
Cuando t = 1 x' = (-6).(1) + 6 = 0
(-1,-5) x - 7 = 0. ( t - 1) Recta x=7
(3,-5) x - 7 = 0. ( t - 1) Recta
= 0° x=7
tg = x'=0
x = (-6)t + 16
RECUERDA, calcula la x que hace tg r = y’ = 0, y habrá un MÁXIMO, si en su entor-no, a su
izquierda, y’= tg r tiene valores positivos, siendo negativos los de la derecha
55. RECAPITULANDO, se calcularán, como ecuación, los valores de x
que hacen que la primera derivada y’ sea nula ( y’=0).
Tendremos un MÁXIMO relativo cuando a la izquierda, de ese
valor, nos de la derivada, del punto elegido, valores positivos, pasando
a negativos, a su derecha.
Tendremos un MÍNIMO relativo cuando a la izquierda, de ese
valor, nos de la derivada, del punto elegido, valores
negativos, pasando a positivos, a su derecha.
EJEMPLO:
y = f (x)= 8x2 – 64 x + 12 su derivada es y’ = f’ (x) = 16x – 64
Para que y’ = 0 16x – 64 = 0 x=4
Valores a la izquierda xiz= 3 f’(xiz)= 16.3 – 64 = - 16
DECRECE
Valores a la derecha xdc= 5 f’(xdc ) = 16.5 – 64 = 16 CRECE
56. En la función y = x3 – 12 x +4 halla los puntos en donde existen máximos y mínimos
Sabiendo que f (x)= y ; f(x +Dx ) = (x +Dx)3 – 12 (x + Dx) +4
f (xx) f ( x) [(x x)3 12( x x) 4] [ x 3 12x 4]
y' f ' ( x) lim x 0 lim x 0
x x
x 3 3x 2 x 3x( x) 2 ( x)3 12x 12 x 4 x3 12x 4
lim x 0
x
3x 2 x 3x( x) 2 ( x)3 12 x x 3x 2 3x x ( x) 2 12
lim x 0 lim x 0
x x
lim x 0 (3x 2 3x x ( x) 2 12) 3x 2 3x.0 02 12 3x 2 12
Para que y’ = 0 3x2 – 12 = 0 Soluciones : x1 = 2 x2= -
2
Valores de la primera derivada en el entorno de x1 = 2
A la izquierda x = 1 f’(1) = 3.12 – 12 = -9 (Negativo) DECRECE
A la derecha x = 3 f’(3) = 3.32 – 12 = 15 (Positivo) CRECE
La tg r varia de negativo a positivo, en el punto x1 = 2 ; y = f(2) =-12 hay un
MÍNIMO
Valores de la primera derivada en el entorno de x2 = -2
A la izquierda x = -3 f’(-3) = 3.(-3)2 – 12 = 15 (Positivo) CRECE
57. En la función x =3t3 – 9t2 + 5 halla los puntos en donde existen máximos y mínimos rela-
tivos.¿Que pasa en esos puntos?
Sabiendo que f (t)= x ; f(t +Dt ) = 3(t +Dt)3 – 9 (t + Dt) +5
f (t t) f (t ) [3(t t ) 3 9(t t ) 2 5] [3t 3 9t 5]
x' f ' (t ) lim t 0 lim t 0
t t
3[t 3 3t 2 t 3t ( t ) 2 ( t ) 3 ] 9[t 2 2t t t 2 ] 5 3t 3 9t 2 5
lim t 0
t
3t 3 9t 2 t 9t ( t ) 2 3( t ) 3 9t 2 18t t 9t 2 5 3t 3 9t 2 5
lim t 0
t
9t 2 t 9t ( t ) 2 3( t ) 3 18t t t (9t 2 9t ( t ) 3( t )2 18t )
lim t 0 lim t 0
t t
lim t 0 lim t 0 9t 2 9t ( t ) 3( t ) 2 18t 9t 2 9t.0 3.(0) 2 18t 9t 2 18t
Haremos que la derivada, la velocidad, se anule, calculando los puntos en que sucede:
f’ (t)=0 9t2 – 18t =0 Soluciones de la Ecuación : t1 = 0 t2 = 2
En el entorno de t1 = 0
A su izquierda, tomamos t =- 1 f’( -1 )= 9. (-1)2 – 18. (-1) = 27 CRECE
A su derecha, tomamos t=1 f’( 1 )= 9. 12 – 18. 1 = -9 DECRECE
Al pasar de Positivo a Negativo en el punto t1 = 0 y1 = 5, hay un MÁXIMO
En el entorno de x2 = 2
A su izquierda, tomamos t=1 f’( 1 )= 9. 12 – 18. 1= - 9 DECRECE
A su derecha, tomamos t= 3 f’( 3 )= 9. 32 – 18. 3 = 27 CRECE
Al pasar de Negativo a Positivo, en el punto t2 = 2 y2 = -7, hay un MÍNIMO
58. Cálculo de derivadas
Si cada vez que tenemos que hallar una derivada hubiese que utilizar el cálculo del
límite, aparte de tedioso, el tiempo perdido seria enorme. Hay que buscar sistemas ( se
llama sistematizar) para resolverla con prontitud y exactitud. Vamos a estudiar los casos
de las derivadas:
59. Derivada de la función constante
y=f(x)=K y+Dy = f ( x + Dx ) =
K
dy f (x x) f ( x) K K 0
y' f ' ( x) lim x 0 lim x 0 lim x 0 0
dx x x x
Es menor el valor cero que el infinitésimo Dx, por eso f’(x) = 0
EJEMPLOS:
y = 1875,6 y’= 0
y = 0’8 y’ = 0
y=p y’ = 0
f(x) = 238,3 y’ = 0
3
f(x) = y’ = 0
4
60. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
y'
61. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
y'
62. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
y' f' x
63. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
y' f' x
64. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy
y' f' x
65. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy
y' f' x
dx
66. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy
y' f' x
dx
67. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy
y' f' x lim
dx x 0
68. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x
y' f' x lim
dx x 0
69. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x
y' f' x lim
dx x 0
70. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x
y' f' x lim
dx x 0
71. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x
y' f' x lim
dx x 0 x
72. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x
y' f' x lim
dx x 0 x
73. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x
y' f' x lim lim
dx x 0 x x 0
74. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x
y' f' x lim lim
dx x 0 x x 0
75. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x
y' f' x lim lim
dx x 0 x x 0
76. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0
77. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
78. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
79. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
80. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
lim
x 0
81. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x
lim
x 0
82. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x
lim
x 0 x
83. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x
lim
x 0 x
84. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim
x 0 x
85. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim
x 0 x 0
86. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim
x 0 x 0
87. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim lim
x 0 x 0 x 0
88. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim lim 1
x 0 x 0 x 0
89. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim lim 1
x 0 x 0 x 0
90. Derivada de la función de 1er Grado
y=f(x)=x y+Dy = f ( x + Dx ) = x + Dx
dy f x x f x x x x
y' f' x lim lim
dx x 0 x x 0 x
x 0
lim lim 1 1
x 0 x 0 x 0
EJEMPLO:
y=x y’ = 1
91. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy
y' f ' ( x)
dx
92. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy
y' f ' ( x) lim x 0
dx
93. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x)
y' f ' ( x) lim x 0
dx
94. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx
95. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
96. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
lim x 0
97. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2
lim x 0
98. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2
lim x 0
99. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2
lim x 0
x
100. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2 x2
lim x 0 lim x 0
x
101. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2 x2 2x x
lim x 0 lim x 0
x
102. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2 x 2 2 x x ( x) 2
lim x 0 lim x 0
x
103. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x
104. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
105. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2x x
lim x 0
106. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2
lim x 0
107. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x 2
lim x 0 lim x 0
x x
2 x x ( x) 2
lim x 0
x
108. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x 2
lim x 0 lim x 0
x x
2 x x ( x) 2 x
lim x 0 lim x 0
x
109. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x 2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(
lim x 0 lim x 0
x
110. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x 2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x
lim x 0 lim x 0
x
111. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x 2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x
112. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x 2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
113. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0
114. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0 2x
115. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0 2x
116. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0 2x x
117. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0 2x x 2x
118. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0 2 x x 2x 0
119. Derivada de la función de 2º Grado
y = f ( x ) = x2 f ( x + Dx ) = ( x + Dx )2
dy f (x x) f ( x)
y ' f ' ( x) lim x 0
dx x
(x x) 2 x 2 x 2 2 x x ( x) 2 x2
lim x 0 lim x 0
x x
2 x x ( x) 2 x(2 x x)
lim x 0 lim x 0
x x
lim x 0 2 x x 2x 0 2x
EJEMPLO: y = x2 y’ = 2x
120. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x)
y' f ' ( x) lim x 0
dx x
121. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x
y' f ' ( x) lim x 0 lim x 0
dx x
122. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x
123. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
124. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
lim x 0
125. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y ' f ' ( x) lim x 0 lim x 0
dx x x
( x x x)
lim x 0
x
126. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y ' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x
lim x 0
x( x x
127. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y ' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x
lim x 0
x( x x
128. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y ' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x)
lim x 0
x( x x x)
129. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2
lim x 0 lim x 0
x( x x x) x( x x x)
130. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
131. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
lim x 0
132. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x
lim x 0
x( x x x)
133. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x
lim x 0
x( x x x)
134. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x
lim x 0
x( x x x)
135. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x x
lim x 0 lim x 0
x( x x x) x( x x x)
136. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x x
lim x 0 lim x 0
x( x x x) x( x x x)
1
lim x 0
x x x
137. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x x
lim x 0 lim x 0
x( x x x) x( x x x)
1 1
lim x 0
x x x x 0 x
138. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x x
lim x 0 lim x 0
x( x x x) x( x x x)
1 1 1
lim x 0
x x x x 0 x x x
139. Derivada de la función Raíz Cuadrada
y f ( x) x y y f (x x) x x
dy f (x x) f ( x) x x x
y' f ' ( x) lim x 0 lim x 0
dx x x
( x x x )( x x x) ( x x )2 ( x )2
lim x 0 lim x 0
x( x x x) x( x x x)
x x x x
lim x 0 lim x 0
x( x x x) x( x x x)
1 1 1 1
lim x 0
x x x x 0 x x x 2 x
140. Derivada de la función de 3º Grado
y = f ( x ) = x3 f ( x + Dx ) = ( x + Dx )3
dy f (x x) f ( x) (x x) 3 x 3 x 3 3 x 2 x 3 x( x) 2 ( x) 3 x3
y ' f ' ( x) lim x 0 lim x 0 lim x 0
dx x x x
3x 2 x 3x( x) 2 ( x) 3 x[3x 2 3x( x) ( x) 2 ]
lim x 0 lim x 0
x x
lim x 0 3x 2 3x( x) ( x) 2 3x 2 3x.0 (0) 2 3x 2
EJEMPLO: y = x3 y’ = 3x2
Derivada de la función Raíz Cúbica
3 3
f ( x) x f (x x) x x
dy f (x x) f ( x) 3
x x 3
x 3
x x 3
x 3
(x x) 2 3
x2 3
x x 3
x
lim x 0 lim x 0 lim x 0
dx x x 3
(x x) 2 3
x2 3
x x 3
x x
3 x x 3
x3 x x x
lim x 0 lim x 0
3 2 3 2 3 3 3 2 3 2 3 3
(x x) x x x x x x (x x) x x x x
x x 1
lim x 0 lim x 0 lim x 0
x 3
(x x) 2 3
x2 3
x x 3
x x 3
(x x) 2 3
x2 3
x x 3
x
1 1 1
1.
3
( x 0) 2 3
x2 3
x 0 3
x 3
x2 3
x2 3
x 3
x 33 x 2
Se trata, también, la función raíz cúbica como una función con exponente 1
3
141. Derivada de la función de Grado n (general)
y = f ( x ) = xn f ( x + Dx ) = ( x + Dx )n
n n n n 1 n n2 n n3 n n
n n
x x x x ( x) 2 x ( x) 3 x
dy f (x x) f ( x) (x x) x 0 1 2 3 n
y' f ' ( x) lim x 0 lim x 0 lim x 0
dx x x x
n.(n 1) n 2
n(n 1)(n 2) n 3 n.(n 1) n 2 n(n 1)(n 2) n 3
xn nxn 1
x x ( x) 2 x ( x) 3 x n nxn 1 x x ( x) 2 x ( x)3
lim 2 2.3 lim x 0 2 2.3
x 0
x x
n.(n 1) n 2 n(n 1)(n 2) n 3
x[nxn 1 x ( x) x ( x) 2 ]
2 2.3 n.(n 1) n 2 n(n 1)(n 2) n 3
lim x 0 lim x 0 nxn 1 x ( x) x ( x) 2
x 2 2.3
n 1 n.(n 1) n 2 n(n 1)(n 2) n 3 n.(n 1) n 2 n(n 1)(n 2) n 3 2
lim x 0 [ nx x ( x) x ( x) 2 ] nxn 1 x .0 x (0) ] n.x n 1
2 2.3 2 2.3
EJEMPLO:
a) y = x8 y’ = 8x8-1=8x7 c) y = x232 y’ = 232x232-1=8x231
b) y = x-5 y’ = -5x-5-1=-5x-6 d)y = x0´3 y’ = 0’3x0’3-1=0’3x-0’7
1 8 8 1 9 8
e) y x y' 8x 8x
x8 x9
5 5 2
7 5 7
5 7 1 5 5
5 5
f) y x x y' x x 2
7 7 5 75 x 2
7x
142. Derivada de la función Raíz e-nésima
1 1 1 n n 1
n n
1 n 1 1 n
1 n
1
f ( x) x x Utilizando la derivada de la potencia e nesim a y ' x x x n 1
n n n n
n.x
1
y'
nn x n 1
1 1 1 8 8 1 7
8 8
1 8 1 1 8
1 8
1 8
1 1
EJEMPLO: y x x y' x x x x 7
8 8 8 8 8 88 x 7
8.x
m m m n n m
n m n
m n 1 m n
m n
m m
y x x suele ser m n y' x x x n m
n n n n nn x n m
n.x
15 15 15 23 8
15 23 1 15 15 1 1 1
EJEMPLO: y 23
x15 x 23
y' x x 23
x 23
8
23 23 23 23 823 x 23 15
823 x8
8.x
143. Derivadas de funciones elementales
¿Como se derivará 8x5 – 3x4 + 8x – 5?. Fijate, ya sabemos como lo debemos hacer
de manera individual,en algunas funciones elementales, ahora tendremos que hacerlo
teniendo en cuenta las conexiones que no son mas que sumas, restas, multiplicaciones
y en su caso divisiones . Por eso vamos a calcular las de las operaciones elementales:
144. Derivada de la Suma de Funciones
y = f ( x ) = g(x) + h ( x f ( x + Dx ) = g (x + Dx) + h (x +
) Dx )
f (xx) f ( x) [ g ( x x) h( x x)] [ g ( x) h( x)]
y' lim x 0 lim x 0
x x
g ( x x ) h( x x ) g ( x ) h( x ) [ g ( x x) g ( x)] [h( x x)] h( x)]
lim x 0 lim x 0
x x
g ( x x) g ( x) h( x x)] h( x) g ( x x) g ( x) h( x x)] h( x)
lim x 0 lim x 0 lim x 0 g ' ( x ) h' ( x )
x x x x
La derivada de una suma de funciones es la suma de las derivadas de cada
función
EJEMPLOS:
a) y = x8 + x5 y’ = 8x7 + 5x4 b) y = x- 3 + x4 y’ = -3x-4 + 4x3
c) y = x-6 + x7 + x y’ =- 6x-7 + 7x6 + 1
d) y = 8 + x-3 + x y’ = 0+(-3)x-4 + 1 =-3x-4 + 1
145. Derivada de la Diferencia de Funciones
y=f(x)=g(x)-h( f ( x + Dx ) = g (x + Dx) – h (x + Dx
x) )
f (xx) f ( x) [ g ( x x) h( x x)] [ g ( x) h( x)]
y' lim x 0 lim x 0
x x
g ( x x ) h( x x ) g ( x ) h( x ) [ g ( x x) g ( x)] [h( x x)] h( x)]
lim x 0 lim x 0
x x
g ( x x) g ( x) h( x x)] h( x) g ( x x) g ( x) h( x x)] h( x)
lim x 0 lim x 0 lim x 0 g ' ( x ) h' ( x )
x x x x
La derivada de una diferencia de funciones es la resta de las derivadas de cada función
EJEMPLOS:
a) y = x9 - x12 y’ = 9x7 - 12x11 b) y = x- 5 – x3 y’ = -5x-6 - 3x2
c) y = x8 - x-7 - 0’35 y’ = 8x7 – (-7)x-8 - 0 = 8x7 + 7x-8
d) y = x122 - x-3 - x y’ = 122x121- (-3)x-4 + 1 = 122x121+3x-4
146. Derivada del Producto de Funciones
y = f ( x ) = g(x) . h ( x ) f ( x + Dx ) = g (x + Dx) . h (x + Dx
f (x x) f ( x) [g(x x).h( x
) x)] [ g ( x).h( x)] g(x x).h( x x) g ( x).h( x)
y' lim x 0 lim x 0 lim x 0
x x x
g(x x).h( x x) g ( x).h( x) g(x x).h( x) g(x x).h( x x)
lim x 0
x
g(x x).h( x x) g ( x).h( x) g(x x).h( x) g(x x).h( x x)
lim x 0
x
g(x x).h( x x) g ( x).h( x) g(x x).h( x x). g(x x).h( x x)
lim x 0
x
g(x x).h( x x) g ( x).h( x x) g ( x).h( x x) g ( x).h( x)
lim x 0
x
g(x x).h( x
x) g ( x).h( x x) g ( x).h( x x) g ( x).h( x)
lim x 0
x x
g(x x).h( x x) g ( x).h( x x) g ( x).h( x x) g ( x).h( x)
lim x 0 lim x 0
x x
h( x x)[g ( x x). g ( x)] g ( x)[h( x x) h( x)] [g(x x). g ( x)]
lim x 0 lim x 0 lim x 0 h( x x).lim x 0
x x x
h( x x ) h( x )
g ( x).lim x 0 h( x 0).g ' ( x) g ( x).h' ( x) h( x).g ' ( x) g ( x).h' ( x)
x
La derivada de un producto de dos funciones es la suma del producto de la
derivada del primer factor por el otro sin derivar mas el de la derivada del segundo
factor por el primero sin derivar
147. Derivada del Producto de Funciones
y’ = f’ ( x ) = g’(x) . h ( x )+ h’(x) . g ( x
)
La derivada de un producto de dos funciones es la suma del producto de la
derivada del primer factor por el otro sin derivar mas el de la derivada del segundo
factor por el primero sin derivar
a) y = 8x5 y’= 0x5 + 8.5x4 =40x4 b) y = px-3 y’= 0x-3 + p.(-3)x-4 = -3px-
4
c) y = 4x2 - 20x - 7 y’= ( 0x2 + 2x) – (0x + 20.1) – 0=2x + 20
d) y = 3x3 + 5x2 - 6 y’= ( 0x3 + 3x2) – (0x2 + 5.2x) – 0= 3x2 + 10x
COROLARIO: La derivada de un producto de una constante por una función es
igual al producto de la constante por la derivada de la función.
y = f ( x ) = K . g(x) y’ = f’ ( x ) = K . g’(x)
y = 25x3 y’= 25.3x2 =75x2 b) y = 0’3x-5 y’= 0’3 .(-5)x-6 = -1’5x-6
148. Derivada del Producto de Funciones
¿ Y CUANDO EL PRODUCTO ES DE MAS DE DOS
FACTORES ?
y = f ( x ) = g(x) . h ( x ). r ( x ) haremos y = f ( x ) = g(x) . [h ( x ). r ( x ) ]
y’ = g’(x) .[ h ( x ). r ( x )]+ g(x) . [h ( x ). r ( x )]’=g’(x) . h ( x ). r ( x )+ g(x) . [h’( x ). r ( x
)]+
La derivada ). r’ un producto. de(varias( funciones. es la ). r ( x del producto x ). la ( x )
g(x) . [h ( x de ( x )]= g’(x) h x ). r x )+ g(x) h’( x suma ) + g(x) . h ( de r’ derivada
del primer factor por los otros, sin derivar, mas el de la derivada del segundo factor por
los otos, sin derivar, mas la del tercero por los otros y así hasta terminar las derivadas
de todos los factores. Sintetizando, es la suma de los productos obtenidos sustituyendo
un factor cualquiera por su derivada.
a) f ( x ) = x . ( x - 1 ). ( x - 8 ) f’ ( x ) = 1. ( x - 1 ). ( x - 8 ) + x . ( 1 - 0 ). ( x - 8 ) + x . ( x - 1 ). ( 1 - 0 )=
= ( x - 1 ). ( x - 8 ) + x . 1 . ( x - 8 ) + x . ( x - 1 ). 1 = ( x - 1 ). ( x - 8 ) + x . ( x - 8 ) + x . ( x - 1 )
b) f ( x ) = 9 x5 . ( x - 12 ). ( x + 3 ) f’ ( x ) = 0 . x5 ( x - 12 ) . ( x + 3 ) + 9.5x4 . ( x - 12 ). ( x + 3
)+
+ 9 x5 . ( 1 - 0 ). ( x + 3 ) + 9 x5 . ( x - 12 ). ( 1 + 0 )= 0 + 45x4 . ( x - 12 ). ( x + 3 ) + 9 x5.( x + 3 ) + 9 x5.( x -
12 )
c) f ( x ) = ( 7 x2 + x + 12 ). ( 3 - 4x -6x2 ). ( x2 - 5 ) f’ ( x ) = ( 14x + 1 + 0 ). ( 3 - 4x -6x2 ). ( x2 - 5 ) +
+ ( 7 x2 + x + 12 ). ( 0 - 4 - 12x ). ( x2 - 5 ) + ( 7 x2 + x + 12 ). ( 3 - 4x -6x2 ).( 2x - 0 )=
2 2 2 2
149. Derivada del Cociente de Funciones
g ( x) g(x x)
y f ( x) y f (x x)
h( x ) h( x x)
g(x x) g ( x) g(x x).h( x ) g ( x ).h( x x)
f (x x) f ( x) h( x x) h( x ) h( x).h( x x)
y ' lim x 0 lim x 0 lim x 0
x x x
g(x x).h( x) g ( x).h( x x) g(x x).h( x) g ( x).h( x x) g ( x).h( x) g ( x).h( x)
lim x 0 lim x 0
h( x).h( x x). x h( x).h( x x ). x
g(x x).h( x) g ( x).h( x) [ g ( x).h( x x) g ( x).h( x)] g(x x).h( x) g ( x).h( x)
lim x 0 lim x 0
h( x).h( x x). x h( x).h( x x). x
g ( x).h( x x) g ( x).h( x) h( x ) g(x x) g ( x)
lim x 0 lim x 0
h( x ).h( x x). x h( x).h( x x) x
g ( x) h( x x ) h( x ) h( x ) g(x x) g ( x)
lim x 0 lim x 0 lim x 0
h( x ).h( x x) x h( x ).h( x x) x
g ( x) h( x x ) h( x ) h( x ) g ( x)
lim x 0 lim x 0 g ' ( x) h' ( x )
h( x ).h( x x) x h( x ).h( x ) h( x).h( x)
g ' ( x).h( x ) g ( x ).h' ( x)
[ h( x )]2
La derivada de un cociente de funciones es la división entre la diferencia de los
productos de la derivada del numerador por el denominador sin derivar menos la
derivada del denominador por el numerador sin derivar partido por el denominador
elevado al cuadrado