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 1E
To determine

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

Expert Solution

Answer to Problem 1E

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

The derivative of y is dydx=43(1+4x)23.

Explanation of Solution

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).

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

F(x)=g(h(x))h(x) (2)

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

g(h(x))=g(u)=ddu(g(u))=ddu(u)13

Apply the power rule (1),

g(h(x))=(13u131)=13(u)133=13(u)23=13(u)23

Substitute u=1+4x in above equation.

g(h(x))=13(1+4x)23

The derivative g(h(x)) is g(h(x))=13(1+4x)23.

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

h(x)=ddx(1+4x)

Apply sum rule (2) and the power rule (1),

h(x)=ddx[1+4x]=ddx[1]+ddx[4x]        =0+4ddx[x]=4[1x11]=4

Thus, the derivative h(x) is h(x)=4.

Substitute 13(1+4x)23 for g(h(x)) and 4 for h(x) in equation (2).

g(h(x))h(x)=13(1+4x)23(4)=43(1+4x)23

Therefore, the derivative of y=1+4x3 is y=43(1+4x)23.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!