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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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

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