Consider the natural logarithm function defined for .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Consider the natural logarithm function

f(x)=\ln x

defined for

x>0

. (a) Calculate

f'(x),\;f''(x),\;f'''(x),\;

and

f^{(4)}(x)

, expressing each derivative in its simplest rational‐function form. (b) Based on the results from part (a), write a general formula for the

n

th derivative

f^{(n)}(x)

, where

n\ge 1

is an integer. Clearly indicate the sign pattern and the power of

x

in the denominator. (c) Explain why the sign of

f^{(n)}(x)

alternates with

n

and describe how the magnitude of

f^{(n)}(x)

behaves as

x\to \infty

. Justify your conclusions without using limits of series.

Related theory

Higher-Order Derivatives
Advanced Differentiation
Second order derivatives
Continuity and Differentiability

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