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 6E
To determine

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

Expert Solution

Answer to Problem 6E

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

The derivative of y is dydx=ex22ex_.

Explanation of Solution

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

Apply the power rule as shown in equation (2)

g(h(x))=12u121=12u122=12u12=12u12

Substitute u=2ex in above equation,

g(h(x))=12(2ex)12=12(2ex)

The derivative g(h(x)) is g(h(x))=12(2ex).

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

h(x)=ddx(2ex)=ddx(2)ddx(ex)=(0)ex=ex

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

Substitute 12(2ex) for g(h(x)) and ex for h(x) in equation (3),

g(h(x))h(x)=12(2ex)(ex)=ex2(2ex)

Therefore, the derivative of y=2ex is dydx=ex22ex_.

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