   Chapter 3.6, Problem 8E

Chapter
Section
Textbook Problem

Differentiate the function. f ( x ) log 10 x

To determine

To find: The derivative of f(x).

Explanation

Given:

The function f(x)=log10x.

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 f(x).

f(x)=ddx(f(x))=ddx(log10x)

Let h(x)=x and g(u)=log10u  where u=h(x)

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

f(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(log10u)

Use the derivative formula ddx(logax)=1xlna,

g(h(x))=1uln10

Substitute u=x in the above equation,

g(h(x))=1xln10

Thus, the derivative g(h(x)) is g(h(x))=1xln10

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