BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.4, Problem 51E
To determine

To find: The value of F(5).

Expert Solution

Answer to Problem 51E

The value of F(5) is F(5)=24.

Explanation of Solution

Given:

The function is F(x)=f(g(x)).

The values are f(2)=8,f(2)=4,f(5)=3,g(5)=2 and g(5)=6.

Result used: Chain Rule

If h is differentiable at x and g is differentiable at h(x), then the composite function F=gh defined by F(x)=g(h(x)) is differentiable at x and F is given by the product

F(x)=g(h(x))h(x) (1)

Calculation:

Obtain the derivative of F(x)=f(g(x)).

F(x)=ddx(F(x))=ddx(f(g(x)))

Apply the chain rule as shown in equation (1)

F(x)=f(g(x))g(x) (2)

Substitute x=5 in equation (2),

F(5)=f(g(5))g(5)

Substitute g(5)=2 in the above equation,

F(5)=f(2)g(5)

Substitute f(2)=4 and g(5)=6 in the above equation,

F(5)=46=24

Therefore, the derivative of F(5)=f(g(5)) is F(5)=24.

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