The document discusses the chain rule, which is used to find the derivative of composite functions. It provides examples of applying the chain rule to functions of the form f(g(x)) by taking the derivative of the outside function with respect to the inside function, and multiplying by the derivative of the inside function with respect to x. The chain rule can be used repeatedly when a function is composed of multiple nested functions. Derivative formulas themselves incorporate the chain rule. The chain rule is essential for finding derivatives and is the most common mistake made by students on tests.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
* Find zeros of polynomial functions
* Use the Fundamental Theorem of Algebra to find a function that satisfies given conditions
* Find all zeros of a polynomial function
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
The following presentation is an introduction to the Algebraic Methods – part one for level 4 Mathematics. This resources is a part of the 2009/2010 Engineering (foundation degree, BEng and HN) courses from University of Wales Newport (course codes H101, H691, H620, HH37 and 001H). This resource is a part of the core modules for the full time 1st year undergraduate programme.
The BEng & Foundation Degrees and HNC/D in Engineering are designed to meet the needs of employers by placing the emphasis on the theoretical, practical and vocational aspects of engineering within the workplace and beyond. Engineering is becoming more high profile, and therefore more in demand as a skill set, in today’s high-tech world. This course has been designed to provide you with knowledge, skills and practical experience encountered in everyday engineering environments.
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Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
1. Made by:
Rudra Patel
Harsh Desai
Ravi Patel
Rishabh Patel
Harshil Raymagiya
2. We now have a pretty good list of “shortcuts” to find
derivatives of simple functions.
Of course, many of the functions that we will encounter
are not so simple. What is needed is a way to combine
derivative rules to evaluate more complicated functions.
®
3. Consider a simple composite function:
y = 6x -10
y = 2( 3x -5)
If u = 3x -5
then y = 2u
y = 6x -10 y = 2u u = 3x - 5
= 2 dy
dy 6
dx
du
= du 3
dx
=
6 = 2×3
dy = dy ×
du
dx du dx
®
4. and another:
y = 5u - 2
where u = 3t
then y = 5( 3t ) - 2
u = 3t
dy 15
dt
= dy 5
du
= du 3
dt
=
15 = 5×3
dy = dy ×
du
dt du dt
y = 5( 3t ) - 2
y =15t - 2
y = 5u - 2
®
5. and one more:
y = 9x2 + 6x +1
( ) 2 y = 3x +1
If u = 3x +1
u = 3x +1
dy 18x 6
dx
y = u2
= + dy 2u
du
= du 3
dx
=
dy = dy ×
du
dx du dx
then y = u2
y = 9x2 + 6x +1
dy 2( 3x 1)
du
= +
dy = 6x +
2
du
18x + 6 = ( 6x + 2) ×3
This pattern is called
the chain rule.
®
6. dy dy du
dx du dx
Chain Rule: = ×
f o g y = f ( u) u = g ( x)
If is the composite of and ,
then:
o ¢ = ¢ × ¢
( ) at u g( x) at x f g f g =
example: f ( x) = sin x g ( x) = x2 - 4 Find: ( f o g )¢ at x = 2
f ¢( x) = cos x g¢( x) = 2x g ( 2) = 4 - 4 = 0
f ¢( 0) × g¢( 2)
cos ( 0) ×( 2×2)
1× 4 = 4
®
7. We could also do it this way:
f ( g ( x) ) = sin ( x2 - 4)
y = sin ( x2 - 4)
y = sin u u = x2 - 4
dy = cosu
du =
2x
du
dx
dy = dy ×
du
dx du dx
dy = cosu ×
2x
dx
dy = cos ( x2 - 4) ×
2x
dx
dy = cos ( 22 - 4) × 2 ×
2
dx
dy = cos( 0) ×
4
dx
dy =
4
dx
®
8. Here is a faster way to find the derivative:
y = sin ( x2 - 4)
y ¢ = cos( x2 - 4) ´ d ( x2 -
4)
dx
y¢ = cos ( x2 - 4) ´2x
Differentiate the outside function...
…then the inside function
At x = 2, y¢ = 4
®
9. Another example:
d cos2 ( 3x)
dx
d éë cos ( 3x
) 2 dx
ùû
2 cos ( 3x) d cos ( 3x)
dx
éë ùû ×
derivative of the
outside function
derivative of the
inside function
It looks like we need to
use the chain rule again!
®
10. Another example:
d cos2 ( 3x)
dx
d éë cos ( 3x
) 2 dx
ùû
2 cos ( 3x) d cos ( 3x)
dx
éë ùû ×
2cos ( 3x) sin ( 3x) d ( 3x)
dx
×- ×
-2cos ( 3x) ×sin ( 3x) ×3
-6cos ( 3x) sin ( 3x)
The chain rule can be used
more than once.
(That’s what makes the
“chain” in the “chain rule”!)
®
11. Derivative formulas include the chain rule!
d un = nun - 1 du
d sin u =
cosu du
dx dx
dx dx
d cosu sin u du
dx dx
= - d tan u =
sec2 u du
dx dx
etcetera…
The formulas on the memorization sheet are written with
instead of . Don’t forget to include the term!
u¢
du u¢
dx
®
12. The most common mistake on the chapter 3 test is to
forget to use the chain rule.
Every derivative problem could be thought of as a chain-rule
problem:
d x2
dx
2x d x
dx
= = 2x ×1 = 2x
derivative of
outside function
derivative of
inside function
The derivative of x is one.
®
13. The chain rule enables us to find the slope of
parametrically defined curves:
dy = dy ×
dx
dt dx dt
dy
dt dy
dx =
dx
dt
Divide both sides by
dx
The slope of a pardatmetrized
curve is given by:
dy
dy dt
dx dx
dt
=
®
14. Example: x = 3cos t y = 2sin t
These are the equations for
an ellipse.
dx 3sin t
dt
= - dy 2cos t
dt
dy t
dx t
= 2cos
=
-
3sin
2 cot
3
= - t
