SlideShare a Scribd company logo
DIFFERENTIAL CALCULUS
BATCH-6:
1. VANATHI.S
2. VARSHINIE.A.L
3. SRIRAM.R
4.THIRUNAVUKARASU.S
5. VISHAL.S
6. SURANJANA.N.S
>>> ENGINEERING MATHEMATICS
TABLE OF CONTENTS
1. INTRODUCTION
2. BAKING ANALOGY
3. DIFFERENTIAL CALCULUS
4. DERIVATIVE RULES
5. PRODUCT RULE
6. QUOTIENT RULE
7. DERIVATIVE OF TRIGNOMETRIC FUCTIONS
8. THE SANDWICH/SQUEEZE THEOREM
9. CONCLUSIONS
INTRODUCTION
>> Differential calculus is a procedure for finding the
exact derivative directly from the formula of the
function without having to use graphical methods.
>> It is a method that deals with the rate of change of
one quantity with respect to another.
BAKING ANALOGY
>> In this we will focus on the formulas and rules for
both differentiation, the method by which we calculate
the derivative of a function.
>> Before we dive into formulas and rules for
differentiation , let’s look at some notations for
differentiation.
>> we can write f(X) as d/dx f(x),f’(x),df(x) and Df(x)
We read these as d by dx of f of x, f’ prime of f of x, df of
x and cap Df of x.
DIFFERENTIAL CALCULUS
>> Differential calculus is the area of calculus dealing
with cutting something into smaller pieces in order to
analyze how it changes.
>> The primary operation of differential calculus is the
derivative. The derivative of a function given the
infinitesimal change of the function with respect to
one of it’s variable.
>> The derivative represents the slope of function at
every point it is defined.
DERIVATIVE RULES
LOGARITHMIC
FUNCTIONS (d/dx):
1. e^x = e^x
2. a^x = a^x ln(a)
3. Ln(x) = 1/x
4. Log a ^x = 1/xln(a)
TRIGONOMETRIC
FUNCTIONS (d/dx):
1. Sin(x)= cos(x)
2. Cos(x)= -sin(x)
3. Tan(x)= sec^2(x)
4. Cosec(x)= - cosec(x) cot(x)
5. Sec(x)= sec(x) tan(x)
6. Cot(x)= -cosec^2(x)
Real life applications of differential
calculus:
>> Calculation of profit or loss with respect to
buissness using graphs.
>> Calculation of rate of change of temperature.
>> To derive many physical equations.
>> Calculation of speed or distance covered such as
miles per hour , km/hour.
PRODUCT RULE
>> The derivative of the product of two differentiable
functions is equal to the addition of the first multiplied
by the derivative of the second and the second
function multiplied by the derivative of the first
function.
APPLICATION:
1. The product rule is used in calculus, when you are
asked to take derivative of the function.
2. It makes calculation clean and easier to solve.
3. It is used to differentiate product of two or more
functions.
DERIVATIVE PRODUCT RULE
If u and v are differentiable at x, then so is their product uv
and
d/dx(u.v) = u (dv/dx) +v (du/dx)
Example: Q) Find the derivative of y=(x^2 +1)(x^3+3)
Answer: d/dx(x^2+1)(x^3+3)=(x^2+1)(3x^2)+ (2x)(x^3+3)
=3x^4+3x^2+2x^4+6x
=5x^4+3x^2+6x
The particular product can be differentiated as well by
multiplying out the original expression for y and
differentiating the resulting polynomial.
Y=(x^2+1)(x^3+3)=x^5+x^3+3x^2+3
dy/dx=5x^4+3x^2+6x
This is in agreement with our first calculation.
QUOTIENT RULE
>> A quotient rule is similar to product rule. A quotient
rule is stated as the ratio of the quantity of the
denominator times the derivative of the numerator
function minus the numerator times the derivative of
the denominator function to the square of the
denominator function.
APPLICATION:
1. It is used for finding the derivative of a quotient of
functions.
2. It is used for extend the power rule to functions with
negative exponents.
3. To combine differentiation rule to find the derivative
of a polynomial or rational function.
DERIVATIVE QUOTIENT RULE
If u and v are differentiable at x and if v(x) is not equal
to 0 then the quotient u/v is differentiable at x and
d/dx(u/v)= v (du/dx) – u (dv/dx)/ v^2
Example: Q)Find the derivative of y=(t^2-1)/(t^3+1)
Answer: u=t^2-1 v=t^3+1
dy/dt=(t^3+1).2t- (t^2-1).3t^2/(t^3+1)^2
=2t^4+2t-3t^4+3t^2/(t^3+1)^2
=-t^4+3t^2+2t/(t^3+1)^2
SQUEEZE THEOREM
>> In calculus the squeeze theorem is a theorem
regarding the limit of a function that is trapped
between two other function.
>> The squeeze theorem is used in calculus and
mathematical analysis typically to confirm the limit of
a function via comparison with other function whose
limits are known.
>> If the right hand limits and left hand limits do not
equal eachother we cannot utilize squeeze theorem.
If f(x)<g(x)<h(x) when x is near a
If limxa f(x)=limxa h(x)=L then limxa g(x)=L.
WHY IS IT CALLED SANDWICH
THEOREM?
>> The squeeze theorem is also called as sandwich or
pinching theorem. It is a way to find the limit of one
function if we know the limits of two functions it is
“sandwiched” between.
APPLICATION:
It is used for calculating the limit of a given
trigonometric funtions.
EXAMPLE OF SANDWICH THEOREM
Q) Using sandwich theorem show that:
limx0 x^2 sin (1/x)=0
ANSWER:
Let -1<sin(1/x)<1
Multiply by x^2
-x^2<x^2 sin 1/x <x^2
Lim x0 (–x^2)<lim x0 x^2 sin (1/x)< lim x0 x^2
Lim x->0 (-x^2)=-0=0
Lim x x^2=0=0
Lim x0 (-x^2)= lim x (x^2)
Lim x0 x^2 sin (1/x)=0
THANK YOU!

More Related Content

Similar to maths diff.calculus ppt (1).pptx

APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...
APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...
APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...
WILIAMMAURICIOCAHUAT1
 
derivativesanditssimpleapplications-160828144729.pptx
derivativesanditssimpleapplications-160828144729.pptxderivativesanditssimpleapplications-160828144729.pptx
derivativesanditssimpleapplications-160828144729.pptx
SnehSinha6
 
Taller parcial 2
Taller parcial 2 Taller parcial 2
Taller parcial 2
XAVIERALEXANDERSALAZ
 
CALCULUS 2.pptx
CALCULUS 2.pptxCALCULUS 2.pptx
CALCULUS 2.pptx
ShienaMaeIndac
 
Aplicaciones de la derivadas en contabilidad
Aplicaciones de la derivadas en contabilidadAplicaciones de la derivadas en contabilidad
Aplicaciones de la derivadas en contabilidad
viniciomuozcONTRERAS
 
CHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudsko
CHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudskoCHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudsko
CHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudsko
SydneyJaydeanKhanyil
 
Multivariate Regression Analysis
Multivariate Regression AnalysisMultivariate Regression Analysis
Multivariate Regression Analysis
Kalaivanan Murthy
 
On the discretized algorithm for optimal proportional control problems constr...
On the discretized algorithm for optimal proportional control problems constr...On the discretized algorithm for optimal proportional control problems constr...
On the discretized algorithm for optimal proportional control problems constr...
Alexander Decker
 
poster2
poster2poster2
poster2
Ryan Grove
 
Grupo#5 taller parcial 2
Grupo#5 taller parcial 2Grupo#5 taller parcial 2
Grupo#5 taller parcial 2
CARLOSANDRESCOLLAGUA
 
Grupo#5 taller parcial 2
Grupo#5 taller parcial 2Grupo#5 taller parcial 2
Grupo#5 taller parcial 2
FernandoSantamara4
 
_lecture_05 F_chain_rule.pdf
_lecture_05 F_chain_rule.pdf_lecture_05 F_chain_rule.pdf
_lecture_05 F_chain_rule.pdf
LeoIrsi
 
Calculus
CalculusCalculus
Calculus
Saira Kanwal
 
Applications of differential equation in Physics and Biology
Applications of differential equation in Physics and BiologyApplications of differential equation in Physics and Biology
Applications of differential equation in Physics and Biology
Ahamed Yoonus S
 
A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...
A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...
A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...
inventionjournals
 
MATH PPT (MC-I)-2.pptx
MATH PPT (MC-I)-2.pptxMATH PPT (MC-I)-2.pptx
MATH PPT (MC-I)-2.pptx
itzsudipto99
 
Controller design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqrController design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqr
eSAT Publishing House
 
Controller design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqrController design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqr
eSAT Journals
 
Ydb ji n8 itc ko6esvj8 kgerx k8tc ko4sx k
Ydb ji n8 itc ko6esvj8 kgerx k8tc ko4sx kYdb ji n8 itc ko6esvj8 kgerx k8tc ko4sx k
Ydb ji n8 itc ko6esvj8 kgerx k8tc ko4sx k
Adikesavaperumal
 
The Fundamental theorem of calculus
The Fundamental theorem of calculus The Fundamental theorem of calculus
The Fundamental theorem of calculus
AhsanIrshad8
 

Similar to maths diff.calculus ppt (1).pptx (20)

APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...
APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...
APLICACIONES DE LA DERIVADA EN LA CARRERA DE (Mecánica, Electrónica, Telecomu...
 
derivativesanditssimpleapplications-160828144729.pptx
derivativesanditssimpleapplications-160828144729.pptxderivativesanditssimpleapplications-160828144729.pptx
derivativesanditssimpleapplications-160828144729.pptx
 
Taller parcial 2
Taller parcial 2 Taller parcial 2
Taller parcial 2
 
CALCULUS 2.pptx
CALCULUS 2.pptxCALCULUS 2.pptx
CALCULUS 2.pptx
 
Aplicaciones de la derivadas en contabilidad
Aplicaciones de la derivadas en contabilidadAplicaciones de la derivadas en contabilidad
Aplicaciones de la derivadas en contabilidad
 
CHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudsko
CHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudskoCHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudsko
CHAPTER 7.pdfdjdjdjdjdjdjdjsjsjddhhdudsko
 
Multivariate Regression Analysis
Multivariate Regression AnalysisMultivariate Regression Analysis
Multivariate Regression Analysis
 
On the discretized algorithm for optimal proportional control problems constr...
On the discretized algorithm for optimal proportional control problems constr...On the discretized algorithm for optimal proportional control problems constr...
On the discretized algorithm for optimal proportional control problems constr...
 
poster2
poster2poster2
poster2
 
Grupo#5 taller parcial 2
Grupo#5 taller parcial 2Grupo#5 taller parcial 2
Grupo#5 taller parcial 2
 
Grupo#5 taller parcial 2
Grupo#5 taller parcial 2Grupo#5 taller parcial 2
Grupo#5 taller parcial 2
 
_lecture_05 F_chain_rule.pdf
_lecture_05 F_chain_rule.pdf_lecture_05 F_chain_rule.pdf
_lecture_05 F_chain_rule.pdf
 
Calculus
CalculusCalculus
Calculus
 
Applications of differential equation in Physics and Biology
Applications of differential equation in Physics and BiologyApplications of differential equation in Physics and Biology
Applications of differential equation in Physics and Biology
 
A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...
A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...
A Study of Some Systems of Linear and Nonlinear Partial Differential Equation...
 
MATH PPT (MC-I)-2.pptx
MATH PPT (MC-I)-2.pptxMATH PPT (MC-I)-2.pptx
MATH PPT (MC-I)-2.pptx
 
Controller design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqrController design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqr
 
Controller design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqrController design of inverted pendulum using pole placement and lqr
Controller design of inverted pendulum using pole placement and lqr
 
Ydb ji n8 itc ko6esvj8 kgerx k8tc ko4sx k
Ydb ji n8 itc ko6esvj8 kgerx k8tc ko4sx kYdb ji n8 itc ko6esvj8 kgerx k8tc ko4sx k
Ydb ji n8 itc ko6esvj8 kgerx k8tc ko4sx k
 
The Fundamental theorem of calculus
The Fundamental theorem of calculus The Fundamental theorem of calculus
The Fundamental theorem of calculus
 

More from Vanathisekar2

Uniform and exponential distribution ppt
Uniform and exponential distribution pptUniform and exponential distribution ppt
Uniform and exponential distribution ppt
Vanathisekar2
 
Varsha.pptx
Varsha.pptxVarsha.pptx
Varsha.pptx
Vanathisekar2
 
mas-150813232504-lva1-app6892.pdf
mas-150813232504-lva1-app6892.pdfmas-150813232504-lva1-app6892.pdf
mas-150813232504-lva1-app6892.pdf
Vanathisekar2
 
evs ppt (2).pptx
evs ppt (2).pptxevs ppt (2).pptx
evs ppt (2).pptx
Vanathisekar2
 
chemistry ppt modified-1.pptx
chemistry ppt modified-1.pptxchemistry ppt modified-1.pptx
chemistry ppt modified-1.pptx
Vanathisekar2
 
E-Textiles.doc
E-Textiles.docE-Textiles.doc
E-Textiles.doc
Vanathisekar2
 
Pspp_Game_development(final).pptx
Pspp_Game_development(final).pptxPspp_Game_development(final).pptx
Pspp_Game_development(final).pptx
Vanathisekar2
 
maths diff.calculus ppt.pptx
maths diff.calculus ppt.pptxmaths diff.calculus ppt.pptx
maths diff.calculus ppt.pptx
Vanathisekar2
 
An Introduction to Metaverse.pdf
An Introduction to Metaverse.pdfAn Introduction to Metaverse.pdf
An Introduction to Metaverse.pdf
Vanathisekar2
 
ranjithreddy123-220304124409.pdf
ranjithreddy123-220304124409.pdfranjithreddy123-220304124409.pdf
ranjithreddy123-220304124409.pdf
Vanathisekar2
 
1.3 Stress & Strain Relationship of Hooke’s Law.ppt
1.3 Stress & Strain Relationship of Hooke’s Law.ppt1.3 Stress & Strain Relationship of Hooke’s Law.ppt
1.3 Stress & Strain Relationship of Hooke’s Law.ppt
Vanathisekar2
 

More from Vanathisekar2 (11)

Uniform and exponential distribution ppt
Uniform and exponential distribution pptUniform and exponential distribution ppt
Uniform and exponential distribution ppt
 
Varsha.pptx
Varsha.pptxVarsha.pptx
Varsha.pptx
 
mas-150813232504-lva1-app6892.pdf
mas-150813232504-lva1-app6892.pdfmas-150813232504-lva1-app6892.pdf
mas-150813232504-lva1-app6892.pdf
 
evs ppt (2).pptx
evs ppt (2).pptxevs ppt (2).pptx
evs ppt (2).pptx
 
chemistry ppt modified-1.pptx
chemistry ppt modified-1.pptxchemistry ppt modified-1.pptx
chemistry ppt modified-1.pptx
 
E-Textiles.doc
E-Textiles.docE-Textiles.doc
E-Textiles.doc
 
Pspp_Game_development(final).pptx
Pspp_Game_development(final).pptxPspp_Game_development(final).pptx
Pspp_Game_development(final).pptx
 
maths diff.calculus ppt.pptx
maths diff.calculus ppt.pptxmaths diff.calculus ppt.pptx
maths diff.calculus ppt.pptx
 
An Introduction to Metaverse.pdf
An Introduction to Metaverse.pdfAn Introduction to Metaverse.pdf
An Introduction to Metaverse.pdf
 
ranjithreddy123-220304124409.pdf
ranjithreddy123-220304124409.pdfranjithreddy123-220304124409.pdf
ranjithreddy123-220304124409.pdf
 
1.3 Stress & Strain Relationship of Hooke’s Law.ppt
1.3 Stress & Strain Relationship of Hooke’s Law.ppt1.3 Stress & Strain Relationship of Hooke’s Law.ppt
1.3 Stress & Strain Relationship of Hooke’s Law.ppt
 

Recently uploaded

Literature review for prompt engineering of ChatGPT.pptx
Literature review for prompt engineering of ChatGPT.pptxLiterature review for prompt engineering of ChatGPT.pptx
Literature review for prompt engineering of ChatGPT.pptx
LokerXu2
 
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...
DharmaBanothu
 
Accident detection system project report.pdf
Accident detection system project report.pdfAccident detection system project report.pdf
Accident detection system project report.pdf
Kamal Acharya
 
Impartiality as per ISO /IEC 17025:2017 Standard
Impartiality as per ISO /IEC 17025:2017 StandardImpartiality as per ISO /IEC 17025:2017 Standard
Impartiality as per ISO /IEC 17025:2017 Standard
MuhammadJazib15
 
一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理
一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理
一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理
nonods
 
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUESAN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
drshikhapandey2022
 
comptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdfcomptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdf
foxlyon
 
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICSUNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
vmspraneeth
 
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
PriyankaKilaniya
 
Call Girls Chennai +91-8824825030 Vip Call Girls Chennai
Call Girls Chennai +91-8824825030 Vip Call Girls ChennaiCall Girls Chennai +91-8824825030 Vip Call Girls Chennai
Call Girls Chennai +91-8824825030 Vip Call Girls Chennai
paraasingh12 #V08
 
FULL STACK PROGRAMMING - Both Front End and Back End
FULL STACK PROGRAMMING - Both Front End and Back EndFULL STACK PROGRAMMING - Both Front End and Back End
FULL STACK PROGRAMMING - Both Front End and Back End
PreethaV16
 
一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理
uqyfuc
 
Open Channel Flow: fluid flow with a free surface
Open Channel Flow: fluid flow with a free surfaceOpen Channel Flow: fluid flow with a free surface
Open Channel Flow: fluid flow with a free surface
Indrajeet sahu
 
Ericsson LTE Throughput Troubleshooting Techniques.ppt
Ericsson LTE Throughput Troubleshooting Techniques.pptEricsson LTE Throughput Troubleshooting Techniques.ppt
Ericsson LTE Throughput Troubleshooting Techniques.ppt
wafawafa52
 
Blood finder application project report (1).pdf
Blood finder application project report (1).pdfBlood finder application project report (1).pdf
Blood finder application project report (1).pdf
Kamal Acharya
 
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Transcat
 
OOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming languageOOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming language
PreethaV16
 
Digital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptxDigital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptx
aryanpankaj78
 
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
ijseajournal
 
Flow Through Pipe: the analysis of fluid flow within pipes
Flow Through Pipe:  the analysis of fluid flow within pipesFlow Through Pipe:  the analysis of fluid flow within pipes
Flow Through Pipe: the analysis of fluid flow within pipes
Indrajeet sahu
 

Recently uploaded (20)

Literature review for prompt engineering of ChatGPT.pptx
Literature review for prompt engineering of ChatGPT.pptxLiterature review for prompt engineering of ChatGPT.pptx
Literature review for prompt engineering of ChatGPT.pptx
 
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...
 
Accident detection system project report.pdf
Accident detection system project report.pdfAccident detection system project report.pdf
Accident detection system project report.pdf
 
Impartiality as per ISO /IEC 17025:2017 Standard
Impartiality as per ISO /IEC 17025:2017 StandardImpartiality as per ISO /IEC 17025:2017 Standard
Impartiality as per ISO /IEC 17025:2017 Standard
 
一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理
一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理
一比一原版(psu学位证书)美国匹兹堡州立大学毕业证如何办理
 
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUESAN INTRODUCTION OF AI & SEARCHING TECHIQUES
AN INTRODUCTION OF AI & SEARCHING TECHIQUES
 
comptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdfcomptia-security-sy0-701-exam-objectives-(5-0).pdf
comptia-security-sy0-701-exam-objectives-(5-0).pdf
 
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICSUNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS
 
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
Prediction of Electrical Energy Efficiency Using Information on Consumer's Ac...
 
Call Girls Chennai +91-8824825030 Vip Call Girls Chennai
Call Girls Chennai +91-8824825030 Vip Call Girls ChennaiCall Girls Chennai +91-8824825030 Vip Call Girls Chennai
Call Girls Chennai +91-8824825030 Vip Call Girls Chennai
 
FULL STACK PROGRAMMING - Both Front End and Back End
FULL STACK PROGRAMMING - Both Front End and Back EndFULL STACK PROGRAMMING - Both Front End and Back End
FULL STACK PROGRAMMING - Both Front End and Back End
 
一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理一比一原版(USF毕业证)旧金山大学毕业证如何办理
一比一原版(USF毕业证)旧金山大学毕业证如何办理
 
Open Channel Flow: fluid flow with a free surface
Open Channel Flow: fluid flow with a free surfaceOpen Channel Flow: fluid flow with a free surface
Open Channel Flow: fluid flow with a free surface
 
Ericsson LTE Throughput Troubleshooting Techniques.ppt
Ericsson LTE Throughput Troubleshooting Techniques.pptEricsson LTE Throughput Troubleshooting Techniques.ppt
Ericsson LTE Throughput Troubleshooting Techniques.ppt
 
Blood finder application project report (1).pdf
Blood finder application project report (1).pdfBlood finder application project report (1).pdf
Blood finder application project report (1).pdf
 
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
Tools & Techniques for Commissioning and Maintaining PV Systems W-Animations ...
 
OOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming languageOOPS_Lab_Manual - programs using C++ programming language
OOPS_Lab_Manual - programs using C++ programming language
 
Digital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptxDigital Twins Computer Networking Paper Presentation.pptx
Digital Twins Computer Networking Paper Presentation.pptx
 
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...Call For Paper -3rd International Conference on Artificial Intelligence Advan...
Call For Paper -3rd International Conference on Artificial Intelligence Advan...
 
Flow Through Pipe: the analysis of fluid flow within pipes
Flow Through Pipe:  the analysis of fluid flow within pipesFlow Through Pipe:  the analysis of fluid flow within pipes
Flow Through Pipe: the analysis of fluid flow within pipes
 

maths diff.calculus ppt (1).pptx

  • 1. DIFFERENTIAL CALCULUS BATCH-6: 1. VANATHI.S 2. VARSHINIE.A.L 3. SRIRAM.R 4.THIRUNAVUKARASU.S 5. VISHAL.S 6. SURANJANA.N.S >>> ENGINEERING MATHEMATICS
  • 2. TABLE OF CONTENTS 1. INTRODUCTION 2. BAKING ANALOGY 3. DIFFERENTIAL CALCULUS 4. DERIVATIVE RULES 5. PRODUCT RULE 6. QUOTIENT RULE 7. DERIVATIVE OF TRIGNOMETRIC FUCTIONS 8. THE SANDWICH/SQUEEZE THEOREM 9. CONCLUSIONS
  • 3. INTRODUCTION >> Differential calculus is a procedure for finding the exact derivative directly from the formula of the function without having to use graphical methods. >> It is a method that deals with the rate of change of one quantity with respect to another.
  • 4. BAKING ANALOGY >> In this we will focus on the formulas and rules for both differentiation, the method by which we calculate the derivative of a function. >> Before we dive into formulas and rules for differentiation , let’s look at some notations for differentiation. >> we can write f(X) as d/dx f(x),f’(x),df(x) and Df(x) We read these as d by dx of f of x, f’ prime of f of x, df of x and cap Df of x.
  • 5. DIFFERENTIAL CALCULUS >> Differential calculus is the area of calculus dealing with cutting something into smaller pieces in order to analyze how it changes. >> The primary operation of differential calculus is the derivative. The derivative of a function given the infinitesimal change of the function with respect to one of it’s variable. >> The derivative represents the slope of function at every point it is defined.
  • 6. DERIVATIVE RULES LOGARITHMIC FUNCTIONS (d/dx): 1. e^x = e^x 2. a^x = a^x ln(a) 3. Ln(x) = 1/x 4. Log a ^x = 1/xln(a) TRIGONOMETRIC FUNCTIONS (d/dx): 1. Sin(x)= cos(x) 2. Cos(x)= -sin(x) 3. Tan(x)= sec^2(x) 4. Cosec(x)= - cosec(x) cot(x) 5. Sec(x)= sec(x) tan(x) 6. Cot(x)= -cosec^2(x)
  • 7. Real life applications of differential calculus: >> Calculation of profit or loss with respect to buissness using graphs. >> Calculation of rate of change of temperature. >> To derive many physical equations. >> Calculation of speed or distance covered such as miles per hour , km/hour.
  • 8. PRODUCT RULE >> The derivative of the product of two differentiable functions is equal to the addition of the first multiplied by the derivative of the second and the second function multiplied by the derivative of the first function. APPLICATION: 1. The product rule is used in calculus, when you are asked to take derivative of the function. 2. It makes calculation clean and easier to solve. 3. It is used to differentiate product of two or more functions.
  • 9. DERIVATIVE PRODUCT RULE If u and v are differentiable at x, then so is their product uv and d/dx(u.v) = u (dv/dx) +v (du/dx) Example: Q) Find the derivative of y=(x^2 +1)(x^3+3) Answer: d/dx(x^2+1)(x^3+3)=(x^2+1)(3x^2)+ (2x)(x^3+3) =3x^4+3x^2+2x^4+6x =5x^4+3x^2+6x The particular product can be differentiated as well by multiplying out the original expression for y and differentiating the resulting polynomial. Y=(x^2+1)(x^3+3)=x^5+x^3+3x^2+3 dy/dx=5x^4+3x^2+6x This is in agreement with our first calculation.
  • 10. QUOTIENT RULE >> A quotient rule is similar to product rule. A quotient rule is stated as the ratio of the quantity of the denominator times the derivative of the numerator function minus the numerator times the derivative of the denominator function to the square of the denominator function. APPLICATION: 1. It is used for finding the derivative of a quotient of functions. 2. It is used for extend the power rule to functions with negative exponents. 3. To combine differentiation rule to find the derivative of a polynomial or rational function.
  • 11. DERIVATIVE QUOTIENT RULE If u and v are differentiable at x and if v(x) is not equal to 0 then the quotient u/v is differentiable at x and d/dx(u/v)= v (du/dx) – u (dv/dx)/ v^2 Example: Q)Find the derivative of y=(t^2-1)/(t^3+1) Answer: u=t^2-1 v=t^3+1 dy/dt=(t^3+1).2t- (t^2-1).3t^2/(t^3+1)^2 =2t^4+2t-3t^4+3t^2/(t^3+1)^2 =-t^4+3t^2+2t/(t^3+1)^2
  • 12. SQUEEZE THEOREM >> In calculus the squeeze theorem is a theorem regarding the limit of a function that is trapped between two other function. >> The squeeze theorem is used in calculus and mathematical analysis typically to confirm the limit of a function via comparison with other function whose limits are known. >> If the right hand limits and left hand limits do not equal eachother we cannot utilize squeeze theorem. If f(x)<g(x)<h(x) when x is near a If limxa f(x)=limxa h(x)=L then limxa g(x)=L.
  • 13. WHY IS IT CALLED SANDWICH THEOREM? >> The squeeze theorem is also called as sandwich or pinching theorem. It is a way to find the limit of one function if we know the limits of two functions it is “sandwiched” between. APPLICATION: It is used for calculating the limit of a given trigonometric funtions.
  • 14. EXAMPLE OF SANDWICH THEOREM Q) Using sandwich theorem show that: limx0 x^2 sin (1/x)=0 ANSWER: Let -1<sin(1/x)<1 Multiply by x^2 -x^2<x^2 sin 1/x <x^2 Lim x0 (–x^2)<lim x0 x^2 sin (1/x)< lim x0 x^2 Lim x->0 (-x^2)=-0=0 Lim x x^2=0=0 Lim x0 (-x^2)= lim x (x^2) Lim x0 x^2 sin (1/x)=0