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

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.4, Problem 5E
To determine

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

Expert Solution

The inner function is u=x and the outer function is f(u)=eu.

The derivative of y is y(x)=ex2x_.

### Explanation of Solution

Given:

The function is y=ex.

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)

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)=x and f(u)=eu. That is,

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

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

Hence, the inner function is u=x and the outer function is f(u)=eu.

Thus, the required form of composite function is .

Obtain the derivative of y is

Let h(x)=x and g(u)=eu  where u=h(x)

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

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

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

g(h(x))=g(u)=ddu(g(u))=ddueu=eu

Substitute u=x in above equation,

g(h(x))=ex

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

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

h(x)=ddx(x)=ddx(x12)

Apply the power rule as shown in equation (2),

h(x)=12x121=12x122=12x12=12x

Thus, the derivative of h(x) is h(x)=12x.

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

g(h(x))h(x)=ex(12x)=ex2x

Therefore, The derivative of y=ex is y(x)=ex2x_.

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