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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Calculate y'.

9. y = ln(x ln x)

To determine

To find: The derivative of y.

Explanation

Given:

The function is y=ln(xlnx).

Result used: Chain Rule

If g is differentiable at t and f is differentiable at g(t), then the composite function F=fg defined by F(t)=f(g(t)) is differentiable at x and F is given by the product

F(t)=f(g(t))g(t) (1)

Derivative Rule:

Product Rule: ddx[f(x)g(x)]=f(x)ddx[g(x)]+g(x)ddx[f(x)]

Calculation:

Obtain the derivative of y.

y=ddx(y)=ddx(ln(xlnx))

Let h(x)=xlnx and f(u)=lnu  where u=h(x).

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

y=f(h(x))h(x) (2)

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

f(h(x))=f(u)            [Qu=h(x)]=ddu

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