   Chapter 3.6, Problem 22E

Chapter
Section
Textbook Problem

Differentiate the function.y = log2 (x log5 x)

To determine

To find: The derivative of y.

Explanation

Given:

The function y=log2(xlog5x).

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 y.

y=ddx(y)=ddx(log2(xlog5x))

Let h(x)=xlog5x and g(u)=log2u  where u=h(x).

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

y=g(h(x))h(x) (2)

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

g(h(x))=g(u)=ddu(g(u))=ddu(log2u)

Use the derivative formula ddx(logax)=1xlna then substitute u=xlog5x,

g(h(x))=1uln2=1(xlog5x)ln2

Thus, the derivative g(h(x)) is g(h(x))=1(xlog5x)ln2

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