   Chapter 3.6, Problem 9E

Chapter
Section
Textbook Problem

Differentiate the function.g(x) = ln(xe–2x)

To determine

To find: The derivative of g(x).

Explanation

Given:

The function g(x)=log(xe2x).

Result used: 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)

Calculation:

Obtain the derivative of g(x).

g(x)=ddx(g(x))=ddx(log(xe2x))

Let h(x)=xe2x and f(u)=logu  where u=h(x)

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

g(x)=f(h(x))h(x) (2)

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

f(h(x))=f(u)=ddu(f(u))=ddu(logu)=1u

Substitute u=xe2x in the above equation,

f(h(x))=1xe2x

Thus, the derivative f(h(x)) is f(h(x))=1xe2x

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