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 2E
To determine

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

Expert Solution

Answer to Problem 2E

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

The derivative of y is dydx=24x2(2x3+5)3.

Explanation of Solution

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)=ddu(g(u))=ddu(u)4

Apply the power rule (2) then substitute u=2x3+5,

g(h(x))=(4u41)=4u3=4(2x3+5)3

The derivative g(h(x)) is g(h(x))=4(2x3+5)3.

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

h(x)=ddx(2x3+5)

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

h(x)=ddx[2x3]+ddx[5]        =2ddx[x3]+[0]=2[3x31]+0=6x2

Thus, the derivative of h(x) is h(x)=6x2.

Substitute 4(2x3+5)3 for g(h(x)) and 6x2 for h(x) in equation (2),

g(h(x))h(x)=4(2x3+5)3(6x2)=24x2(2x3+5)3

Therefore, The derivative of y=(2x3+5)4 is dydx=24x2(2x3+5)3.

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