   Chapter 3.4, Problem 2E

Chapter
Section
Textbook Problem

Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx.y = (2x3 + 5)4

To determine

To find: The composite function in the form f(g(x)) and obtain the derivative of y.

Explanation

Given:

The function is y=(2x3+5)4.

Formula used:

The 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)

Derivative Rule:

(1) Power Rule:ddx(xn)=nxn1.

(2) Sum Rule: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

Calculation:

Let the inner function be u=g(x) and the outer function be y=f(u).

Then, g(x)=2x3+5 and f(u)=u4. That is,

y=(2x3+5)4=f(2x3+5)=f(g(x))

Therefore, y=f(g(x)).

Hence, the inner function is u=2x3+5 and the outer function is f(u)=u4.

Thus, the required form of composite function is f(g(x))=u4.

Obtain the derivative of y .

Let h(x)=2x3+5 and g(u)=(u)4  where u=h(x)

Apply the chain rule as shown in equation (1)

y(x)=g(h(x))h(x) (2)

The derivative g(h(x)) is computed as follows,

g(h(x))=g(u<

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