Download The Derivative of Composite Functions: Understanding the Chain Rule and more Exams Advanced Calculus in PDF only on Docsity!
NEL
Section 2.5 —The Derivatives of Composite
Functions
Recall that one way of combining functions is through a process called composition. We start with a number x in the domain of g , find its image and then take the value of f at provided that is in the domain of f. The result is the new function which is called the composite function of f and g, and is denoted.
EXAMPLE 1 Reflecting on the process of composition
If and find each of the following values:
a. b. c. d.
Solution a. Since we have b. Since we have Note:. c. d. Note:.
The chain rule states how to compute the derivative of the composite function h 1 x 2 f 1 g 1 x 22 in terms of the derivatives of f and g.
g 1 f 1 x 22 g 1 V x 2 V x 5 f 1 g 1 x 22 g 1 f 1 x 22
f 1 g 1 x 22 f 1 x 52 V x 5
f 142 2, g 1 f 1422 g 122 7. f 1 g 1422 g 1 f 1422
g 142 9, f 1 g 1422 f 192 3.
f 1 g 1422 g 1 f 1422 f 1 g 1 x 22 g 1 f 1 x 22
f 1 x 2 V x g 1 x 2 x 5,
1 f g 2
h 1 x 2 f 1 g 1 x 22 ,
g 1 x 2 , g 1 x 2
g 1 x 2 ,
Definition of a composite function Given two functions f and g , the composite function is defined by 1 f g 2 1 x 2 f 1 g 1 x 22.
1 f g 2
The Chain Rule If f and g are functions that have derivatives, then the composite function h 1 x 2 f 1 g 1 x 22 has a derivative given by h ¿ 1 x 2 f ¿ 1 g 1 x 22 g ¿ 1 x 2.
C H A P T E R 2 (^) 99
In words, the chain rule says, “the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.”
100 2. 5^ T H E D E R I VAT I V E S O F C O M P O S I T E F U N C T I O N S NEL
The Chain Rule in Leibniz Notation
If y is a function of u and u is a function of x (so that y is a composite function), then provided that and dudx exist.
dy du
dy dx ^
dy du
du dx ,
Proof: By the definition of the derivative, Assuming that we can write
(Property of limits)
Since
Therefore,
This proof is not valid for all circumstances. When dividing by we assume that A proof that covers all cases can be found in advanced calculus textbooks.
EXAMPLE 2 Applying the chain rule Differentiate
Solution The inner function is , and the outer function is
The derivative of the inner function is
The derivative of the outer function is
The derivative of the outer function evaluated at the inner function is
By the chain rule, h ¿ 1 x 2 32 1 x^2 x 2
(^12) 12 x 12.
f ¿ 1 x^2 x 2 32 1 x^2 x 2
(^12) .
g 1 x 2
f ¿ 1 x 2 32 x
(^12) .
g ¿ 1 x 2 2 x 1.
f 1 x 2 x
(^32) g 1 x 2 x^2 x.
h 1 x 2 1 x^2 x 2
(^32) .
g 1 x h 2 g 1 x 2 0.
g 1 x h 2 g 1 x 2 ,
3 f 1 g 1 x 22 4 ¿ f ¿ 1 g 1 x 22 g ¿ 1 x 2.
3 f 1 g 1 x 22 4 ¿ lim k S 0
c
f 1 g 1 x 2 k 2 f 1 g 1 x 22 k
d lim h S 0 c
g 1 x h 2 g 1 x 2 h
d
as h S 0. We obtain
lim h S 0
3 g 1 x h 2 g 1 x 2 4 0, let g 1 x h 2 g 1 x 2 k and k S 0
lim h S 0
c
f 1 g 1 x h 22 f 1 g 1 x 22 g 1 x h 2 g 1 x 2
d lim c h S 0
g 1 x h 2 g 1 x 2 h
d
3 f 1 g 1 x 22 4 ¿ lim h S 0
c a
f 1 g 1 x h 22 f 1 g 1 x 22 g 1 x h 2 g 1 x 2
b a
g 1 x h 2 g 1 x 2 h
b d
g 1 x h 2 g 1 x 2 0,
3 f 1 g 1 x 22 4 ¿ lim h S 0
f 1 g 1 x h 22 f 1 g 1 x 22 h.
If we interpret derivatives as rates of change, the chain rule states that if y is a function of x through the intermediate variable u , then the rate of change of y
102 2. 5^ T H E D E R I VAT I V E S O F C O M P O S I T E F U N C T I O N S NEL
EXAMPLE 6 Connecting the derivative to the slope of a tangent Using a graphing calculator, sketch the graph of the function
Find the equation of the tangent at the point on the graph.
Solution Using a graphing calculator, the graph is
The slope of the tangent at point is given by We first write the function as By the power of a function rule, The slope at is
The equation of the tangent is y 1 ^121 x 22 , or x 2 y 4 0.
1 2, 1 2 f ¿ 122 814 42 ^2142
f ¿ 1 x 2 81 x^2 42 ^212 x 2.
f 1 x 2 81 x^2 42 ^1.
1 2, 1 2 f ¿ 122.
f 1 x 2 (^) x (^2 8) 4.
EXAMPLE 5 Using the chain rule to differentiate a power of a function If
Solution The inner function is and the outer function is By the chain rule,
Example 5 is a special case of the chain rule in which the outer function is a power function of the form This leads to a generalization of the power rule seen earlier.
y 3 g 1 x 2 4 n.
14 x 1 x^2 526
dy dx
71 x^2 52612 x 2
g 1 x 2 x^2 5, f 1 x 2 x^7.
y 1 x^2 527 , find dydx.
Power of a Function Rule If n is a real number and then ,
or (^) dxd 3 g 1 x 2 4 n^ n 3 g 1 x 2 4 n ^1 g ¿ 1 x 2.
d dx^1 u^
n (^) 2 nu n 1 du u g 1 x 2 , dx
For help using the graphing calculator to graph functions and draw tangent lines see Technical Appendices p. 597 and p. 608.
Tech Support
NEL
EXAMPLE 7 Combining derivative rules to differentiate a complex product
Differentiate Express your answer
in a simplified factored form.
Solution Here we use the product rule and the chain rule.
(Product rule)
(Chain rule) (Simplify) (Factor)
EXAMPLE 8 Combining derivative rules to differentiate a complex quotient
Determine the derivative of Solution A – Using the product and chain rule There are several approaches to this problem. You could keep the function as it is and use the chain rule and the quotient rule. You could also decompose the function and express it as and then apply the quotient rule and the chain rule. Here we will express the function as the product and apply the product rule and the chain rule.
(Simplify) (Factor)
(Rewrite using positive exponents)
Solution B – Using the chain and quotient rule In this solution, we will use the chain rule and the quotient rule, where is the inner function and is the outer function.
g ¿ 1 x 2
dg du
du dx
u 1 ^ x u^10
2 1 x^2
40 x 11 x^2 11 x^2
20 x 11 x^2 2911 x^2 2 ^11122
20 x 11 x^2 2911 x^2 2 ^11 3 1 1 x^2 2 11 x^2 2 4
20 x 11 x^2 2911 x^2 2 ^10 120 x 2 1 1 x^2 21011 x^2 2 ^11
1011 x^2 2912 x 2 1 1 x^2 2 ^10 11 x^2 2101 10 2 1 1 x^2 2 ^111 2 x 2
g ¿ 1 x 2
d dx
c 11 x^2 210 d 11 x^2 2 ^10 11 x^2
d dx
c 11 x^2 2 ^10 d
g 1 x 2 11 x^2 21011 x^2 2 ^10
g 1 x 2
11 x^2 11 x^2 210 ,
g 1 x 2 Q^1 ^ x
2 1 x^2 R
10 .
41 x^2 32314 x 522111 x^2 10 x 92
41 x^2 32314 x 522 32 x 14 x 52 31 x^2 3 2 4
8 x 1 x^2 323 14 x 523 1214 x 5221 x^2 324
341 x^2 32312 x 2 4 14 x 523 3314 x 52214 2 4 1 x^2 324
h ¿ 1 x 2
d dx
3 1 x^2 324 4 14 x 523
d dx
3 1 4 x 523 4 1 x^2 324
h 1 x 2 1 x^2 32414 x 523.
C H A P T E R 2 (^) 103
NEL
Exercise 2.
PART A
- Given and find the following value: a. c. e. b. d. f.
- For each of the following pairs of functions, find the composite functions and. What is the domain of each composite function? Are the composite functions equal?
a. b. c.
- What is the rule for calculating the derivative of the composition of two differentiable functions? Give examples, and show how the derivative is determined.
- Differentiate each function. Do not expand any expression before differentiating. a. d. b. e.
c. f.
PART B
- Rewrite each of the following in the form or and then differentiate.
a. c. e.
b. d. f.
- Given where f and g are continuous functions, use the information in the table to evaluate and
- Given f 1 x 2 1 x 322 , g 1 x 2 (^1) x , and h 1 x 2 f 1 g 1 x 22 ,determine h 1 x 2.
h 1 12 h 1 12.
h g f ,
y
1 x^2 x 124
y
9 x^2
y
x 1
y
5 x^2 x
y
x^2 4
y
x^3
y u n y ku n ,
f 1 x 2
1 x^2 1625
h 1 x 2 12 x^2 3 x 524
g 1 x 2 1 x^2 423 y V x^2 3
f 1 x 2 12 x 324 f 1 x 2 1 p^2 x^2
g 1 x 2 V x g 1 x 2 x^2 1 g 1 x 2 V x 2
f 1 x 2
x
f 1 x 2
x
f 1 x 2 x^2
1 f g 2 1 g f 2
g 1 f 1122 f 1 g 1 422 g 1 f 1 x 22
f 1 g 1122 g 1 f 1022 f 1 g 1 x 22
f 1 x 2 V x g 1 x 2 x^2 1,
K
C H A P T E R 2 (^) 105
C
x f ( x ) g ( x ) f ’( x ) g ’( x ) 1 1 18 ^5 ^15 0 2 5 1 11 1 1 4 3 7 2 4 9 7 3 3 13 10 11 1
106 2. 5^ T H E D E R I VAT I V E S O F C O M P O S I T E F U N C T I O N S NEL
- Differentiate each function. Express your answer in a simplified factored form. a. d. b. e.
c. f.
- Find the rate of change of each function at the given value of t. Leave your answers as rational numbers, or in terms of roots and the number.
a. b. s ( t ) =
- For what values of x do the curves and have the same slope?
- Find the slope of the tangent to the curve at
- Find the equation of the tangent to the curve at
- Use the chain rule, in Leibniz notation, to find at the given value of x. a. b. c. d.
- Find , given and
- A 50 000 L tank can be drained in 30 min. The volume of water remaining in the tank after t minutes is At what rate, to the nearest whole number, is the water flowing out of the tank when?
- The function represents the displacement s , in metres, of a particle moving along a straight line after t seconds. Determine the velocity when.
PART C
- a. Write an expression for if b. If find
- Show that the tangent to the curve at the point is also the tangent to the curve at another point.
- Differentiate y x
(^211) x (^3 ) 11 x 2 3.
y 1 x^2 x 223 3 1 1, 3 2
h 1 x 2 x 12 x 7241 x 122 , h ¿ 1 32.
h ¿ 1 x 2 h 1 x 2 p 1 x 2 q 1 x 2 r 1 x 2.
t 3
s 1 t 2 1 t^3 t^2
(^12) , t 0,
t 10
V 1 t 2 50 000 Q 1 30 t R 2 , 0 t 30.
h ¿ 122 h 1 x 2 f 1 g 1 x 22 , f 1 u 2 u^2 1, g 122 3, g ¿ 122 1.
y u^3 51 u^3 7 u 22 , u V x , x 4
y u 1 u^2 323 , u 1 x 322 , x 2
y 2 u^3 3 u^2 , u x x
(^12) , x 1
y 3 u^2 5 u 2, u x^2 1, x 2
dy dx
y 1 x^3 725 x 2.
y 13 x x^2 2 ^2 Q2, 14 R.
y 11 x^3 22 y 2 x^6
a t 2 p
t p t 6 p
b
(^13) s 1 t 2 t ,
(^13) 14 t 52
(^23) , t 8
p
y a
x^2 3 x^2 3
b
4 y
3 x^2 2 x x^2 1
y 1 x^2 3231 x^3 322 y x^411 4 x^2
f 1 x 2 1 x 4231 x 326 h 1 x 2 x^313 x 522
A
T
108 C A R E E R L I N K W R A P - U P NEL
CAREER LINK WRAP-UP Investigate and Apply
CHAPTER 2: THE ELASTICITY OF DEMAND
An electronics retailing chain has established the monthly price –demand relationship for an electronic game as
They are trying to set a price level that will provide maximum revenue ( R ). They know that when demand is elastic a drop in price will result in higher overall revenues and that when demand is inelastic an increase in price will result in higher overall revenues. To complete the questions in this task, you will have to use the elasticity definition
converted into differential notation. a. Determine the elasticity of demand at $20 and $80, classifying these price points as having elastic or inelastic demand. What does this say about where the optimum price is in terms of generating the maximum revenue? Explain. Also calculate the revenue at the $20 and $80 price points. b. Approximate the demand curve as a linear function (tangent) at a price point of $50. Plot the demand function and its linear approximation on the graphing calculator. What do you notice? Explain this by looking at the demand function. c. Use your linear approximation to determine the price point that will generate the maximum revenue. ( Hint: Think about the specific value of E where you will not want to increase or decrease the price to generate higher revenues.) What revenue is generated at this price point? d. A second game has a price–demand relationship of
The price is currently set at $50. Should the company increase or decrease the price? Explain.
n 1 p 2
12 500 p 25
Q¢ ¢ np dndp R
E c a
¢ n n
b a
¢ p p
b d
1 R np 2 , 1 E 6 12 ,
1 E 7 12 ,
n 1 p 2 1000 10
1 p 12
(^43)
^3 p
1 p 2 1 n 2
