By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)

Size: px
Start display at page:

Download "By writing (1) as y (x 5 1). (x 5 1), we can find the derivative using the Product Rule: y (x 5 1) 2. we know this from (2)"

Transcription

1 3.5 Chain Rule Chain Rule Introuction As iscusse in Section 3.2, the Power Rule is vali for all real number exponents n. In this section we see that a similar rule hols for the erivative of a power of a function y [g(x)] n. Before stating the formal result, let us consier an example when n is a positive integer. Suppose we wish to ifferentiate By writing (1) as y (x 5 1). (x 5 1), we can fin the erivative using the Prouct Rule: 2(x 5 1). 5x 4. Similarly, to ifferentiate the function y (x 5 1) 3, we can write it as y (x 5 1) 2. (x 5 1) an use the Prouct Rule an the result given in (2): 3(x 5 1) 2. 5x 4. x x n nx n1 y (x 5 1) 2. x (x5 1) 2 (x 5 1). x (x5 1) (x 5 1). x (x5 1) (x 5 1). 5x 4 (x 5 1). 5x 4 x (x5 1) 3 x (x5 1) 2. (x 5 1) we know this from (2) (x 5 1) 2. x (x5 1) (x 5 1). x (x5 1) 2 (x 5 1) 2. 5x 4 (x 5 1). 2(x 5 1). 5x 4 In like manner, by writing y (x 5 1) 4 as y (x 5 1) 3. (x 5 1) we can reaily show by the Prouct Rule an (3) that x (x5 1) 4 4(x 5 1) 3. 5x 4. Power Rule for Functions Inspection of (2), (3), an (4) reveals a pattern for ifferentiating a power of a function g. For example, in (4) we see bring own exponent as a multiple erivative of function insie parentheses 4(x 5 1) 3. 5x 4 c ecrease exponent by 1 For emphasis, if we enote a ifferentiable function by x [ ]n n[ ] n1 x [ ]. [ ], it appears that he foregoing iscussion suggests the result state in the next theorem. (1) (2) (3) (4) heorem Power Rule for Functions If n is any real number an u g(x) is ifferentiable at x, then x [g(x)]n n[g(x)] n1. g (x), or equivalently, (6) x un nu n1. u x. (5)

2 150 CHAPER 3 he Derivative heorem is itself a special case of a more general theorem, calle the Chain Rule, which will be presente after we consier some examples of this new power rule. EXAMPLE 1 Power Rule for Functions Differentiate y (4x 3 3x 1) 7. Solution With the ientification that u g(x) 4x 3 3x 1, we see from (6) that n u n1 u>x y x 7(4x3 3x 1) 6. x (4x3 3x 1) 7(4x 3 3x 1) 6 (12x 2 3). { EXAMPLE 2 Power Rule for Functions o ifferentiate y 1>(x 2 1), we coul, of course, use the Quotient Rule. However, by rewriting the function as y (x 2 1) 1, it is also possible to use the Power Rule for Functions with n 1: EXAMPLE 3 y x (1)(x2 1) 2. x (x2 1) (1)(x 2 1) 2 2x 2x (x 2 1) 2. Power Rule for Functions 1 Differentiate y (7x 5 x 4 2) 10. Solution Write the given function as y (7x 5 x 4 2) 10. Ientify u 7x 5 x 4 2, n 10 an use the Power Rule (6): y x 10(7x5 x 4 2) 11. x (7x5 x 4 2) 10(35x4 4x 3 ) (7x 5 x 4 2) 11. EXAMPLE 4 Power Rule for Functions Differentiate y tan 3 x. Solution For emphasis, we first rewrite the function as y (tan x) 3 an then use (6) with u tan x an n 3: y x 3(tan x)2. tan x. x Recall from (6) of Section 3.4 that (>x)tan x sec 2 x. Hence, y x 3 tan2 x sec 2 x. EXAMPLE 5 Quotient Rule then Power Rule Differentiate y (x2 1) 3 (5x 1) 8. Solution We start with the Quotient Rule followe by two applications of the Power Rule for Functions: Power Rule for Functions y (5x 1)8. x x (x2 1) 3 (x 2 1) 3. (5x 1)8 x (5x 1) 16 (5x 1)8. 3(x 2 1) 2. 2x (x 2 1) 3. 8(5x 1) 7. 5 (5x 1) 16

3 3.5 Chain Rule 151 6x(5x 1)8 (x 2 1) 2 40(5x 1) 7 (x 2 1) 3 (5x 1) 16 (x2 1) 2 (10x 2 6x 40) (5x 1) 9. EXAMPLE 6 Differentiate Solution Power Rule then Quotient Rule y A 2x 3 8x 1. By rewriting the function as 2x 3 y Q 8x 1 R 1>2 we can ientify an n 1 2. hus in orer to compute u>x in (6) we must use the Quotient Rule: y x x 1 R 1>2. 3 x 8x 1 R x 1 R 1>2. (8x 1). 2 (2x 3). 8 (8x 1) x 1 R 1>2. 26 (8x 1) 2. Finally, we simplify using the laws of exponents: u y x 13 (2x 3) 1>2 (8x 1) 3>2. 2x 3 8x 1 Chain Rule A power of a function can be written as a composite function. If we ientify f (x) x n an u g(x), then f (u) f (g(x)) [g(x)] n. he Chain Rule gives us a way of ifferentiating any composition f g of two ifferentiable functions f an g. heorem Chain Rule If the function f is ifferentiable at u g(x), an the function g is ifferentiable at x, then the composition y ( f g)(x) f (g(x)) is ifferentiable at x an x f (g(x)) f (g(x)). g (x) y y or equivalently, (8) x u. u x. (7) PROOF FOR u 0 In this partial proof it is convenient to use the form of the efinition of the erivative given in (3) of Section 3.1. For x 0, u g(x x) g(x) or g(x x) g(x) u u u. In aition, y f (u u) f (u) f (g(x x)) f (g(x)). When x an x x are in some open interval for which u 0, we can write y y x u. u x. (9)

4 152 CHAPER 3 he Derivative Since g is assume to be ifferentiable, it is continuous. Consequently, as x S 0, g(x x) S g(x), an so from (9) we see that u S 0. hus, From the efinition of the erivative, (3) of Section 3.1, it follows that he assumption that u 0 on some interval oes not hol true for every ifferentiable function g. Although the result given in (7) remains vali when u 0, the preceing proof oes not. It might help in the unerstaning of the erivative of a composition y f (g(x)) to think of f as the outsie function an u g(x) as the insie function. he erivative of y f (g(x)) f (u) is then the prouct of the erivative of the outsie function (evaluate at the insie function u) an the erivative of the insie function (evaluate at x): erivative of outsie function x f (u) f (u). u. c erivative of insie function he result in (10) is written in various ways. Since y f (u), we have f (u) y>u, an of course u u>x. he prouct of the erivatives in (10) is the same as (8). On the other han, if we replace the symbols u an u in (10) by g(x) an g (x) we obtain (7). Proof of the Power Rule for Functions As note previously, a power of a function can be written as a composition of ( f g)(x) where the outsie function is y f (x) x n an the y n1 insie function is u g(x). he erivative of the insie function y f (u) u n is nu x u an the erivative of the outsie function is he prouct of these erivatives is then x. y y x u. u x nu n1 u x n[g(x)]n1 g (x). his is the Power Rule for Functions given in (5) an (6). rigonometric Functions We obtain the erivatives of the trigonometric functions compose with a ifferentiable function g as another irect consequence of the Chain Rule. For example, if y sin u, where u g(x), then the erivative of y with respect to the variable u is Hence, (8) gives or equivalently, y y lim Q lim x xs0 u R. u Q lim xs0 x R xs0 y Q lim us0 u R. u Q lim xs0 x R. y y x u. u x. y cos u. u y y x u. u x cos u u x. x sin[ ] cos[ ] x [ ] Similarly, if y tan u where u g(x), then y>u sec 2 u an so y y x u. u x u sec2 u x. note that u S 0 in the first term We summarize the Chain Rule results for the six trigonometric functions. (10)

5 3.5 Chain Rule 153 heorem Derivatives of rigonometric Functions If u g(x) is a ifferentiable function, then x sin u cos u u x, x tan u u sec2 u x, x secu sec u tan u u x, x cos u sin u u x, x cot u u csc2 u x, x csc u csc u cot u u x. (11) (12) (13) EXAMPLE 7 Chain Rule Differentiate y cos 4x. Solution he function is cos u with u 4x. From the secon formula in (11) of heorem the erivative is y u u x y x sin 4x. 4x 4 sin 4x. x EXAMPLE 8 Chain Rule Differentiate y tan(6x 2 1). Solution he function is tan u with u 6x 2 1. From the first formula in (12) of heorem the erivative is u sec 2 u x y x sec2 (6x 2 1). x (6x2 1) 12x sec 2 (6x 2 1). EXAMPLE 9 Prouct, Power, an Chain Rule Differentiate y (9x 3 1) 2 sin 5x. Solution We first use the Prouct Rule: y x (9x3 1) 2. x sin 5x sin 5x. x (9x3 1) 2 followe by the Power Rule (6) an the first formula in (11) of heorem 3.5.3, from (11) from (6) y x (9x3 1) 2. cos 5x. x 5x sin 5x. 2(9x 3 1). (9x 3 1) 2. 5 cos 5x sin 5x. 2(9x 3 1). x (9x3 1) 27x 2 (9x 3 1)(45x 3 cos 5x 5 cos 5x 54x 2 sin 5x). In Sections 3.2 an 3.3 we saw that even though the Sum an Prouct Rules were state in terms of two functions f an g, they were applicable to any finite number of ifferentiable functions. So too, the Chain Rule is state for the composition of two functions f an g but we can apply it to the composition of three (or more) ifferentiable functions. In the case of three functions f, g, an h, (7) becomes x f (g(h(x))) f (g(h(x))). x g(h(x)) f (g(h(x))). g (h(x)). h (x).

6 154 CHAPER 3 he Derivative EXAMPLE 10 Repeate Use of the Chain Rule Differentiate y cos 4 (7x 3 6x 1). Solution For emphasis we first rewrite the given function as y [cos(7x 3 6x 1)] 4. Observe that this function is the composition ( f g h)(x) f (g(h(x))) where f (x) x 4, g(x) cosx, an h(x) 7x 3 6x 1. We first apply the Chain Rule in the form of the Power Rule (6) followe by the secon formula in (11): y x 4[cos (7x3 6x 1)] 3. x cos (7x3 6x 1) 4 cos 3 (7x 3 6x 1). csin(7x 3 6x 1). x (7x3 6x 1) 4(21x 2 6) cos 3 (7x 3 6x 1) sin (7x 3 6x 1). In the final example, the given function is a composition of four functions. first Chain Rule: ifferentiate the power secon Chain Rule: ifferentiate the cosine EXAMPLE 11 Differentiate Repeate Use of the Chain Rule y sin( tan23x 2 4). Solution he function is f (g(h(k(x)))), where f (x) sin x, g(x) tan x, h(x) 1x, an k(x) 3x 2 4. In this case we apply the Chain Rule three times in succession y x cos Atan23x 2 4 B. x tan23x2 4 cos Atan23x 2 4 B. sec 2 23x 2 4. x 23x2 4 cos Atan23x 2 4 B. sec 2 23x 2 4. x (3x2 4) 1>2 cos Atan23x 2 4 B. sec 2 23x (3x2 4) 1>2. x (3x2 4) first Chain Rule: ifferentiate the sine secon Chain Rule: ifferentiate the tangent rewrite power thir Chain Rule: ifferentiate the power cos Atan23x 2 4 B. sec 2 23x (3x2 4) 1>2. 6x 3x cos Atan23x2 4 B. sec 2 23x x 2 4 simplify You shoul, of course, become so aept at applying the Chain Rule that you will not have to give a moment s thought as to the number of functions involve in the actual composition. x NOES FROM HE CLASSROOM (i) Probably the most common mistake is to forget to carry out the secon half of the Chain Rule, namely the erivative of the insie function. his is the u>x part in y y u x u x. 56 For instance, the erivative of y (1 x) 57 is not y>x 57(1 x) since 57(1 x) 56 is only the y>u part. It might help to consistently use the operation symbol >x: x (1 x) 57 57(1 x) 56. x (1 x) 57(1 x) 56. (1).

7 3.5 Chain Rule 155 (ii) A less common but probably a worse mistake than the first is to ifferentiate insie the given function. A stuent wrote on an examination paper that the erivative of y cos (x 2 1) was y>x sin (2x); that is, the erivative of the cosine is the negative of the sine an the erivative of x 2 1 is 2x. Both observations are correct, but how they are put together is incorrect. Bear in min that the erivative of the insie function is a multiple of the erivative of the outsie function. Again, it might help to use the operation symbol >x. he correct erivative of y cos (x 2 1) is the prouct of two erivatives. y x sin (x 2 1). x (x 2 1) 2x sin (x 2 1).

( 3x +1) 2 does not fit the requirement of the power rule that the base be x

( 3x +1) 2 does not fit the requirement of the power rule that the base be x Section 3 4A: The Chain Rule Introuction The Power Rule is state as an x raise to a real number If y = x n where n is a real number then y = n x n-1 What if we wante to fin the erivative of a variable

More information

February 21 Math 1190 sec. 63 Spring 2017

February 21 Math 1190 sec. 63 Spring 2017 February 21 Math 1190 sec. 63 Spring 2017 Chapter 2: Derivatives Let s recall the efinitions an erivative rules we have so far: Let s assume that y = f (x) is a function with c in it s omain. The erivative

More information

2.5 The Chain Rule Brian E. Veitch

2.5 The Chain Rule Brian E. Veitch 2.5 The Chain Rule This is our last ifferentiation rule for this course. It s also one of the most use. The best way to memorize this (along with the other rules) is just by practicing until you can o

More information

Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis

Math Chapter 2 Essentials of Calculus by James Stewart Prepared by Jason Gaddis Math 231 - Chapter 2 Essentials of Calculus by James Stewart Prepare by Jason Gais Chapter 2 - Derivatives 21 - Derivatives an Rates of Change Definition A tangent to a curve is a line that intersects

More information

Define each term or concept.

Define each term or concept. Chapter Differentiation Course Number Section.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan

More information

The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier.

The Chain Rule. Composition Review. Intuition. = 2(1.5) = 3 times faster than (X)avier. The Chain Rule In the previous section we ha to use a trig ientity to etermine the erivative of. h(x) = sin(2x). We can view h(x) as the composition of two functions. Let g(x) = 2x an f (x) = sin x. Then

More information

d dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0.

d dx [xn ] = nx n 1. (1) dy dx = 4x4 1 = 4x 3. Theorem 1.3 (Derivative of a constant function). If f(x) = k and k is a constant, then f (x) = 0. Calculus refresher Disclaimer: I claim no original content on this ocument, which is mostly a summary-rewrite of what any stanar college calculus book offers. (Here I ve use Calculus by Dennis Zill.) I

More information

Differentiation Rules Derivatives of Polynomials and Exponential Functions

Differentiation Rules Derivatives of Polynomials and Exponential Functions Derivatives of Polynomials an Exponential Functions Differentiation Rules Derivatives of Polynomials an Exponential Functions Let s start with the simplest of all functions, the constant function f(x)

More information

Differentiability, Computing Derivatives, Trig Review. Goals:

Differentiability, Computing Derivatives, Trig Review. Goals: Secants vs. Derivatives - Unit #3 : Goals: Differentiability, Computing Derivatives, Trig Review Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an

More information

Using the definition of the derivative of a function is quite tedious. f (x + h) f (x)

Using the definition of the derivative of a function is quite tedious. f (x + h) f (x) Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit

More information

1 Lecture 18: The chain rule

1 Lecture 18: The chain rule 1 Lecture 18: The chain rule 1.1 Outline Comparing the graphs of sin(x) an sin(2x). The chain rule. The erivative of a x. Some examples. 1.2 Comparing the graphs of sin(x) an sin(2x) We graph f(x) = sin(x)

More information

Review of Differentiation and Integration for Ordinary Differential Equations

Review of Differentiation and Integration for Ordinary Differential Equations Schreyer Fall 208 Review of Differentiation an Integration for Orinary Differential Equations In this course you will be expecte to be able to ifferentiate an integrate quickly an accurately. Many stuents

More information

1 Definition of the derivative

1 Definition of the derivative Math 20A - Calculus by Jon Rogawski Chapter 3 - Differentiation Prepare by Jason Gais Definition of the erivative Remark.. Recall our iscussion of tangent lines from way back. We now rephrase this in terms

More information

Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions

Section The Chain Rule and Implicit Differentiation with Application on Derivative of Logarithm Functions Section 3.4-3.6 The Chain Rule an Implicit Differentiation with Application on Derivative of Logarithm Functions Ruipeng Shen September 3r, 5th Ruipeng Shen MATH 1ZA3 September 3r, 5th 1 / 3 The Chain

More information

1 Lecture 20: Implicit differentiation

1 Lecture 20: Implicit differentiation Lecture 20: Implicit ifferentiation. Outline The technique of implicit ifferentiation Tangent lines to a circle Derivatives of inverse functions by implicit ifferentiation Examples.2 Implicit ifferentiation

More information

f(x) f(a) Limit definition of the at a point in slope notation.

f(x) f(a) Limit definition of the at a point in slope notation. Lesson 9: Orinary Derivatives Review Hanout Reference: Brigg s Calculus: Early Transcenentals, Secon Eition Topics: Chapter 3: Derivatives, p. 126-235 Definition. Limit Definition of Derivatives at a point

More information

Math 251 Notes. Part I.

Math 251 Notes. Part I. Math 251 Notes. Part I. F. Patricia Meina May 6, 2013 Growth Moel.Consumer price inex. [Problem 20, page 172] The U.S. consumer price inex (CPI) measures the cost of living base on a value of 100 in the

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Implicit Differentiation Using the Chain Rule In the previous section we focuse on the erivatives of composites an saw that THEOREM 20 (Chain Rule) Suppose that u = g(x) is ifferentiable

More information

Differentiability, Computing Derivatives, Trig Review

Differentiability, Computing Derivatives, Trig Review Unit #3 : Differentiability, Computing Derivatives, Trig Review Goals: Determine when a function is ifferentiable at a point Relate the erivative graph to the the graph of an original function Compute

More information

Section 2.1 The Derivative and the Tangent Line Problem

Section 2.1 The Derivative and the Tangent Line Problem Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan

More information

Tutorial 1 Differentiation

Tutorial 1 Differentiation Tutorial 1 Differentiation What is Calculus? Calculus 微積分 Differential calculus Differentiation 微分 y lim 0 f f The relation of very small changes of ifferent quantities f f y y Integral calculus Integration

More information

Chapter 2 Derivatives

Chapter 2 Derivatives Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,

More information

Section 7.1: Integration by Parts

Section 7.1: Integration by Parts Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises

More information

A. Incorrect! The letter t does not appear in the expression of the given integral

A. Incorrect! The letter t does not appear in the expression of the given integral AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question

More information

Math 115 Section 018 Course Note

Math 115 Section 018 Course Note Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of

More information

TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12

TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12 NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus

More information

2.1 Derivatives and Rates of Change

2.1 Derivatives and Rates of Change 1a 1b 2.1 Derivatives an Rates of Change Tangent Lines Example. Consier y f x x 2 0 2 x-, 0 4 y-, f(x) axes, curve C Consier a smooth curve C. A line tangent to C at a point P both intersects C at P an

More information

Lecture 6: Calculus. In Song Kim. September 7, 2011

Lecture 6: Calculus. In Song Kim. September 7, 2011 Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear

More information

Math 190 Chapter 3 Lecture Notes. Professor Miguel Ornelas

Math 190 Chapter 3 Lecture Notes. Professor Miguel Ornelas Math 190 Chapter 3 Lecture Notes Professor Miguel Ornelas 1 M. Ornelas Math 190 Lecture Notes Section 3.1 Section 3.1 Derivatives of Polynomials an Exponential Functions Derivative of a Constant Function

More information

MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points

MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points MA4001 Engineering Mathematics 1 Lecture 14 Derivatives of Trigonometric Functions Critical Points Dr. Sarah Mitchell Autumn 2014 An important limit To calculate the limits of basic trigonometric functions

More information

Chapter 3 Definitions and Theorems

Chapter 3 Definitions and Theorems Chapter 3 Definitions an Theorems (from 3.1) Definition of Tangent Line with slope of m If f is efine on an open interval containing c an the limit Δy lim Δx 0 Δx = lim f (c + Δx) f (c) = m Δx 0 Δx exists,

More information

Differentiation ( , 9.5)

Differentiation ( , 9.5) Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the

More information

Chapter 1 Overview: Review of Derivatives

Chapter 1 Overview: Review of Derivatives Chapter Overview: Review of Derivatives The purpose of this chapter is to review the how of ifferentiation. We will review all the erivative rules learne last year in PreCalculus. In the net several chapters,

More information

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1 Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of

More information

QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim

QF101: Quantitative Finance September 5, Week 3: Derivatives. Facilitator: Christopher Ting AY 2017/2018. f ( x + ) f(x) f(x) = lim QF101: Quantitative Finance September 5, 2017 Week 3: Derivatives Facilitator: Christopher Ting AY 2017/2018 I recoil with ismay an horror at this lamentable plague of functions which o not have erivatives.

More information

UNDERSTANDING INTEGRATION

UNDERSTANDING INTEGRATION UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,

More information

Rules of Differentiation. Lecture 12. Product and Quotient Rules.

Rules of Differentiation. Lecture 12. Product and Quotient Rules. Rules of Differentiation. Lecture 12. Prouct an Quotient Rules. We warne earlier that we can not calculate the erivative of a prouct as the prouct of the erivatives. It is easy to see that this is so.

More information

Integration Review. May 11, 2013

Integration Review. May 11, 2013 Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In

More information

Derivatives and the Product Rule

Derivatives and the Product Rule Derivatives an the Prouct Rule James K. Peterson Department of Biological Sciences an Department of Mathematical Sciences Clemson University January 28, 2014 Outline Differentiability Simple Derivatives

More information

Computing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL

Computing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL Computing Derivatives by J. Douglas Chil, Ph.D. Rollins College Winter Park, FL ii Computing Inefinite Integrals Important notice regaring book materials Texas Instruments makes no warranty, either express

More information

Math 1 Lecture 20. Dartmouth College. Wednesday

Math 1 Lecture 20. Dartmouth College. Wednesday Math 1 Lecture 20 Dartmouth College Wenesay 10-26-16 Contents Reminers/Announcements Last Time Derivatives of Trigonometric Functions Reminers/Announcements WebWork ue Friay x-hour problem session rop

More information

Hyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures

Hyperbolic Functions. Notice: this material must not be used as a substitute for attending. the lectures Hyperbolic Functions Notice: this material must not be use as a substitute for attening the lectures 0. Hyperbolic functions sinh an cosh The hyperbolic functions sinh (pronounce shine ) an cosh are efine

More information

Chapter 3 Notes, Applied Calculus, Tan

Chapter 3 Notes, Applied Calculus, Tan Contents 3.1 Basic Rules of Differentiation.............................. 2 3.2 The Prouct an Quotient Rules............................ 6 3.3 The Chain Rule...................................... 9 3.4

More information

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like

Math Implicit Differentiation. We have discovered (and proved) formulas for finding derivatives of functions like Math 400 3.5 Implicit Differentiation Name We have iscovere (an prove) formulas for fining erivatives of functions like f x x 3x 4x. 3 This amounts to fining y for 3 y x 3x 4x. Notice that in this case,

More information

1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) =

1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) = Theorem 13 (i) If p(x) is a polynomial, then p(x) = p(c) 1 Limits 11 12 Fining its graphically Examples 1 f(x) = x3 1, x 1 x 1 The behavior of f(x) as x approximates 1 x 1 f(x) = 3 x 2 f(x) = x+1 1 f(x)

More information

Outline. MS121: IT Mathematics. Differentiation Rules for Differentiation: Part 1. Outline. Dublin City University 4 The Quotient Rule

Outline. MS121: IT Mathematics. Differentiation Rules for Differentiation: Part 1. Outline. Dublin City University 4 The Quotient Rule MS2: IT Mathematics Differentiation Rules for Differentiation: Part John Carroll School of Mathematical Sciences Dublin City University Pattern Observe You may have notice the following pattern when we

More information

THEOREM: THE CONSTANT RULE

THEOREM: THE CONSTANT RULE MATH /MYERS/ALL FORMULAS ON THIS REVIEW MUST BE MEMORIZED! DERIVATIVE REVIEW THEOREM: THE CONSTANT RULE The erivative of a constant function is zero. That is, if c is a real number, then c 0 Eample 1:

More information

Final Exam Study Guide and Practice Problems Solutions

Final Exam Study Guide and Practice Problems Solutions Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making

More information

0.1 The Chain Rule. db dt = db

0.1 The Chain Rule. db dt = db 0. The Chain Rule A basic illustration of the chain rules comes in thinking about runners in a race. Suppose two brothers, Mark an Brian, hol an annual race to see who is the fastest. Last year Mark won

More information

Implicit Differentiation and Inverse Trigonometric Functions

Implicit Differentiation and Inverse Trigonometric Functions Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5

More information

Chapter 2. Exponential and Log functions. Contents

Chapter 2. Exponential and Log functions. Contents Chapter. Exponential an Log functions This material is in Chapter 6 of Anton Calculus. The basic iea here is mainly to a to the list of functions we know about (for calculus) an the ones we will stu all

More information

Breakout Session 13 Solutions

Breakout Session 13 Solutions Problem True or False: If f = 2, then f = 2 False Any time that you have a function of raise to a function of, in orer to compute the erivative you nee to use logarithmic ifferentiation or something equivalent

More information

With the Chain Rule. y x 2 1. and. with respect to second axle. dy du du dx. Rate of change of first axle. with respect to third axle

With the Chain Rule. y x 2 1. and. with respect to second axle. dy du du dx. Rate of change of first axle. with respect to third axle 0 CHAPTER Differentiation Section The Chain Rule Fin the erivative of a composite function using the Chain Rule Fin the erivative of a function using the General Power Rule Simplif the erivative of a function

More information

Calculus in the AP Physics C Course The Derivative

Calculus in the AP Physics C Course The Derivative Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.

More information

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

Math 1271 Solutions for Fall 2005 Final Exam

Math 1271 Solutions for Fall 2005 Final Exam Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly

More information

MAT137 Calculus! Lecture 6

MAT137 Calculus! Lecture 6 MAT137 Calculus! Lecture 6 Today: 3.2 Differentiation Rules; 3.3 Derivatives of higher order. 3.4 Related rates 3.5 Chain Rule 3.6 Derivative of Trig. Functions Next: 3.7 Implicit Differentiation 4.10

More information

Solutions to Practice Problems Tuesday, October 28, 2008

Solutions to Practice Problems Tuesday, October 28, 2008 Solutions to Practice Problems Tuesay, October 28, 2008 1. The graph of the function f is shown below. Figure 1: The graph of f(x) What is x 1 + f(x)? What is x 1 f(x)? An oes x 1 f(x) exist? If so, what

More information

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever

More information

Mathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY

Mathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY Mathematics 1210 PRACTICE EXAM II Fall 2018 ANSWER KEY 1. Calculate the following: a. 2 x, x(t) = A sin(ωt φ) t2 Solution: Using the chain rule, we have x (t) = A cos(ωt φ)ω = ωa cos(ωt φ) x (t) = ω 2

More information

Differentiation Rules and Formulas

Differentiation Rules and Formulas Differentiation Rules an Formulas Professor D. Olles December 1, 01 1 Te Definition of te Derivative Consier a function y = f(x) tat is continuous on te interval a, b]. Ten, te slope of te secant line

More information

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule

Unit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin

More information

102 Problems Calculus AB Students Should Know: Solutions. 18. product rule d. 19. d sin x. 20. chain rule d e 3x2) = e 3x2 ( 6x) = 6xe 3x2

102 Problems Calculus AB Students Should Know: Solutions. 18. product rule d. 19. d sin x. 20. chain rule d e 3x2) = e 3x2 ( 6x) = 6xe 3x2 Problems Calculus AB Stuents Shoul Know: Solutions. + ) = + =. chain rule ) e = e = e. ) =. ) = ln.. + + ) = + = = +. ln ) =. ) log ) =. sin ) = cos. cos ) = sin. tan ) = sec. cot ) = csc. sec ) = sec

More information

Calculus I Announcements

Calculus I Announcements Slie 1 Calculus I Announcements Office Hours: Amos Eaton 309, Monays 12:50-2:50 Exam 2 is Thursay, October 22n. The stuy guie is now on the course web page. Start stuying now, an make a plan to succee.

More information

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0.

x f(x) x f(x) approaching 1 approaching 0.5 approaching 1 approaching 0. Engineering Mathematics 2 26 February 2014 Limits of functions Consier the function 1 f() = 1. The omain of this function is R + \ {1}. The function is not efine at 1. What happens when is close to 1?

More information

Computing Derivatives Solutions

Computing Derivatives Solutions Stuent Stuy Session Solutions We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions to account for groups that are able to answer the questions at a faster rate. Use

More information

Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review)

Math 1A Midterm 2 Fall 2015 Riverside City College (Use this as a Review) Name Date Miterm Score Overall Grae Math A Miterm 2 Fall 205 Riversie City College (Use this as a Review) Instructions: All work is to be shown, legible, simplifie an answers are to be boxe in the space

More information

MATH2231-Differentiation (2)

MATH2231-Differentiation (2) -Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha

More information

1 Lecture 13: The derivative as a function.

1 Lecture 13: The derivative as a function. 1 Lecture 13: Te erivative as a function. 1.1 Outline Definition of te erivative as a function. efinitions of ifferentiability. Power rule, erivative te exponential function Derivative of a sum an a multiple

More information

Derivatives of Trigonometric Functions

Derivatives of Trigonometric Functions Derivatives of Trigonometric Functions 9-8-28 In this section, I ll iscuss its an erivatives of trig functions. I ll look at an important it rule first, because I ll use it in computing the erivative of

More information

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes

3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we

More information

Math 1B, lecture 8: Integration by parts

Math 1B, lecture 8: Integration by parts Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores

More information

SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)

SESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive) SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first

More information

DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS

DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS DERIVATIVES: LAWS OF DIFFERENTIATION MR. VELAZQUEZ AP CALCULUS THE DERIVATIVE AS A FUNCTION f x = lim h 0 f x + h f(x) h Last class we examine the limit of the ifference quotient at a specific x as h 0,

More information

Recapitulation of Mathematics

Recapitulation of Mathematics Unit I Recapitulation of Mathematics Basics of Differentiation Rolle s an Lagrange s Theorem Tangent an Normal Inefinite an Definite Integral Engineering Mathematics I Basics of Differentiation CHAPTER

More information

Integration by Parts

Integration by Parts Integration by Parts 6-3-207 If u an v are functions of, the Prouct Rule says that (uv) = uv +vu Integrate both sies: (uv) = uv = uv + u v + uv = uv vu, vu v u, I ve written u an v as shorthan for u an

More information

Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut

Antiderivatives. Definition (Antiderivative) If F (x) = f (x) we call F an antiderivative of f. Alan H. SteinUniversity of Connecticut Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Antierivatives Definition (Antierivative) If F (x) = f (x) we call F an antierivative of f. Definition (Inefinite

More information

CHAPTER 3 DERIVATIVES (continued)

CHAPTER 3 DERIVATIVES (continued) CHAPTER 3 DERIVATIVES (continue) 3.3. RULES FOR DIFFERENTIATION A. The erivative of a constant is zero: [c] = 0 B. The Power Rule: [n ] = n (n-1) C. The Constant Multiple Rule: [c *f()] = c * f () D. The

More information

Linear and quadratic approximation

Linear and quadratic approximation Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function

More information

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:

cosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities: 6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan

More information

SYDE 112, LECTURE 1: Review & Antidifferentiation

SYDE 112, LECTURE 1: Review & Antidifferentiation SYDE 112, LECTURE 1: Review & Antiifferentiation 1 Course Information For a etaile breakown of the course content an available resources, see the Course Outline. Other relevant information for this section

More information

The Chain Rule. y x 2 1 y sin x. and. Rate of change of first axle. with respect to second axle. dy du. du dx. Rate of change of first axle

The Chain Rule. y x 2 1 y sin x. and. Rate of change of first axle. with respect to second axle. dy du. du dx. Rate of change of first axle . The Chain Rule 9. The Chain Rule Fin the erivative of a composite function using the Chain Rule. Fin the erivative of a function using the General Power Rule. Simplif the erivative of a function using

More information

Differential and Integral Calculus

Differential and Integral Calculus School of science an engineering El Akhawayn University Monay, March 31 st, 2008 Outline 1 Definition of hyperbolic functions: The hyperbolic cosine an the hyperbolic sine of the real number x are enote

More information

Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x

Math 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x . Fin erivatives of the following functions: (a) f() = tan ( 2 + ) ( ) 2 (b) f() = ln 2 + (c) f() = sin() Solution: Math 80, Eam 2, Fall 202 Problem Solution (a) The erivative is compute using the Chain

More information

Section 7.2. The Calculus of Complex Functions

Section 7.2. The Calculus of Complex Functions Section 7.2 The Calculus of Complex Functions In this section we will iscuss limits, continuity, ifferentiation, Taylor series in the context of functions which take on complex values. Moreover, we will

More information

Chapter 5 Analytic Trigonometry

Chapter 5 Analytic Trigonometry Chapter 5 Analytic Trigonometry Section 1 Section 2 Section 3 Section 4 Section 5 Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas

More information

Fall 2016: Calculus I Final

Fall 2016: Calculus I Final Answer the questions in the spaces provie on the question sheets. If you run out of room for an answer, continue on the back of the page. NO calculators or other electronic evices, books or notes are allowe

More information

Lecture 5: Inverse Trigonometric Functions

Lecture 5: Inverse Trigonometric Functions Lecture 5: Inverse Trigonometric Functions 5 The inverse sine function The function f(x = sin(x is not one-to-one on (,, but is on [ π, π Moreover, f still has range [, when restricte to this interval

More information

4.2 First Differentiation Rules; Leibniz Notation

4.2 First Differentiation Rules; Leibniz Notation .. FIRST DIFFERENTIATION RULES; LEIBNIZ NOTATION 307. First Differentiation Rules; Leibniz Notation In this section we erive rules which let us quickly compute the erivative function f (x) for any polynomial

More information

Summary: Differentiation

Summary: Differentiation Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin

More information

MAT01A1: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules

MAT01A1: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules MAT01A1: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 22 Marc 2017 Semester Test 1 Scripts will be available for collection from Tursay morning. For marking

More information

Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval

Exam 3 Review. Lesson 19: Concavity, Inflection Points, and the Second Derivative Test. Lesson 20: Absolute Extrema on an Interval Exam 3 Review Lessons 17-18: Relative Extrema, Critical Numbers, an First Derivative Test (from exam 2 review neee for curve sketching) Critical Numbers: where the erivative of a function is zero or unefine.

More information

Inverse Trig Functions

Inverse Trig Functions 6.6i Inverse Trigonometric Functions Inverse Sine Function Does g(x) = sin(x) have an inverse? What restriction would we need to make so that at least a piece of this function has an inverse? Given f (x)

More information

Chapter Primer on Differentiation

Chapter Primer on Differentiation Capter 0.01 Primer on Differentiation After reaing tis capter, you soul be able to: 1. unerstan te basics of ifferentiation,. relate te slopes of te secant line an tangent line to te erivative of a function,.

More information

You should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises.

You should also review L Hôpital s Rule, section 3.6; follow the homework link above for exercises. BEFORE You Begin Calculus II If it has been awhile since you ha Calculus, I strongly suggest that you refresh both your ifferentiation an integration skills. I woul also like to remin you that in Calculus,

More information

Further Differentiation and Applications

Further Differentiation and Applications Avance Higher Notes (Unit ) Prerequisites: Inverse function property; prouct, quotient an chain rules; inflexion points. Maths Applications: Concavity; ifferentiability. Real-Worl Applications: Particle

More information

The Exact Form and General Integrating Factors

The Exact Form and General Integrating Factors 7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily

More information

Integration: Using the chain rule in reverse

Integration: Using the chain rule in reverse Mathematics Learning Centre Integration: Using the chain rule in reverse Mary Barnes c 999 University of Syney Mathematics Learning Centre, University of Syney Using the Chain Rule in Reverse Recall that

More information