   Chapter 3.4, Problem 6E

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 = 2 − e x

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=2ex.

Result 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)

Power Rule:

If n is positive integer, then ddx(xn)=nxn1 (2)

Calculation:

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

Then, g(x)=2ex and f(u)=u. That is,

y=2ex=f(2ex)=f(g(x))

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

Hence, the inner function is u=2ex and the outer function is f(u)=u.

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

Obtain the derivative of y.

y=ddx(y)=ddx(2ex)

Let h(x)=2ex and g(u)=u where u=h(x)

Apply the chain rule as shown in equation (1),

y(x)=g(h(x))h(x) (3)

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

g(h(x))=g(u)=ddu(g(u))=dduu=ddu(u)12

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

Evaluate the expression sin Exercises 116. 22212424

Finite Mathematics and Applied Calculus (MindTap Course List)

find the real roots of each equation by factoring. 129. x2 + x 12 = 0

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

The area of the shaded region is given by:

Study Guide for Stewart's Multivariable Calculus, 8th

The length of the are r = e for 0 is given by: a) 0ed b) 02e2d c) 02ed d) 02ed

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