   Chapter 3.4, Problem 1E

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 = 1 + 4 x 3

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=1+4x3.

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)=1+4x and f(u)=u3. That is,

y=1+4x3=f(1+4x)=f(g(x))

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

Hence, the inner function is u=1+4x and the outer function is f(u)=u3.

Thus, the required form of composite function is f(g(x))=u3.

Obtain the derivative of y.

y=ddx(y)=ddx(1+4x3)

Let h(x)=1+4x and g(u)=(u)13  where u=h(x)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

If f is even, then f is even.

Single Variable Calculus: Early Transcendentals, Volume I

In Problems 33 – 38, solve each inequality. 38.

Mathematical Applications for the Management, Life, and Social Sciences

The average value of f(x) = 3x2 + 1 on the interval [2, 4] is: a) 29 b) 66 c) 58 d) 36

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 