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Problems on Functions and Derivatives by The Chain Rule - Assignment | MS 125, Assignments of Calculus

Material Type: Assignment; Professor: Kim; Class: Calculus I; Subject: Mathematics (MS); University: Jacksonville State University; Term: Fall 2006;

Typology: Assignments

Pre 2010

Uploaded on 08/16/2009

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J. Kim MS 125 Worksheet 3.4 (Sep. 29, 2006)
Section 3.4 The Chain Rule
Example 1 Express each function as a composition of two functions.
1.
5
2
34)( xxF
2.
13
)(
x
exG
3.
253)(
2
xxxH
4.
4ln)(
2
xxT
The Chain Rule
If
)(xf
and
)(xg
are differentiable functions, then the composition function
)(xgf
the two
functions is differentiable, and the derivative of is
)()()( xgxgfxgf
dx
d
In words, the derivative of a composite function is the product of the derivatives of the outside and
inside functions. The derivative of the outside function must be evaluated at the inside function.
Example 2 Find the derivatives of the following functions by using the chain rule.
1.
2.
253)(
2
xxxg
3.
24
1
)( xx
xh
4.
35
)(
x
exf
5.
2
)(
x
exg
6.
5)(
2
x
exh
7.
19
2
1)(
t
exf
Example 3 Find the derivatives of the following functions by using appropriate rules and formulas.
1.
143
)(
x
exxf
2.
1
1
)(
3
x
x
e
e
xh

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J. Kim MS 125 Worksheet 3. 4 (Sep. 29, 2006)

Section 3. 4 The Chain Rule

Example 1 Express each function as a composition of two functions.

2 5 F ( x ) 4 x  3

G ( x )  e 3 x ^1

3. H ( x ) 3 x^2  5 x  2

4. T (^ x )ln^ ^ x^2 ^4 

The Chain Rule

If f^ ( x ) and g^ ( x )are differentiable functions, then the composition function ^ f^ ^ g ^ ( x )the two

functions is differentiable, and the derivative of is

 f  g ( x ) f  g ( x ) g ( x )

dx d   

In words, the derivative of a composite function is the product of the derivatives of the outside and

inside functions. The derivative of the outside function must be evaluated at the inside function.

Example 2 Find the derivatives of the following functions by using the chain rule.

1. f ( x )( x^2  1 )^10

2. g ( x ) 3 x^2  5 x  2

x x

h x

f ( x )  e 5 x ^3

2 g ( x )  ex

6. h ( x ) e ^2 x  5

2 19 f ( x ) 1  e t

Example 3 Find the derivatives of the following functions by using appropriate rules and formulas.

f ( x )  x^3 e ^4 x ^1

3

 x

x

e

e

h x