   Chapter 3.6, Problem 53E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find a formula for f(n)(x) if f(x) = ln(x – 1).

To determine

To find: The formula for fn(x).

Explanation

Given:

The function f(x)=ln(x1).

Result used: Chain Rule

If y=f(u) and u=g(x) are both differentiable functions, then dydx=dydududx.

(1) Quotient Rule: ddx(fg)=gddx(f)fddx(g)(g)2 .

Calculation:

The first derivative of f(x)=ln(x1) is computed as follows,

f(x)=ddx(f(x))=ddx(ln(x1))

Let u=x1 and apply the chain rule as stated above,

f(x)=ddx(lnu)=1ududx=1x1ddx(x1)     (Qu=x1)=1x1(10)

=1x1

Thus, the first derivative of f(x)=ln(x1) is f(x)=1x1_.

The second derivative of f(x)=ln(x1) is computed as follows,

f(x)=ddx(f(x))=ddx(1x1)

Apply the quotient rule (1),

f(x)=(x1)ddx(1)(1)ddx(x1)(x1)2=(x1)(0)(10)(x1)2=01(x1)2=1(x1)2

Thus, the second derivative of f(x)=ln(x1) is f(x)=1(x1)2_

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 