Derivative of Logarithmic function

Formula

$\dfrac{d}{dx}{\, \log_{e}{f{(x)}}}$ $\,=\,$ ${\dfrac{1}{f{(x)}}}{\dfrac{d}{dx}{\, f{(x)}}}$

The derivative of logarithm of a function is equal to the product of the reciprocal of the function and the derivative of the function.

Introduction

$f{(x)}$ is a function in terms of $x$ and its natural logarithm is written as $\log_{e}{f{(x)}}$ or $\ln{f{(x)}}$. The differentiation of the natural logarithm of the function $f{(x)}$ with respect to $x$ is written mathematically as follows.

$\dfrac{d}{dx}{\, \log_{e}{[f{(x)}]}}$

The derivative of the natural logarithm of function $f{(x)}$ with respect to $x$ is equal to the product of the reciprocal of the function $f{(x)}$ and derivative of the function $f{(x)}$ with respect to $x$.

$\dfrac{d}{dx}{\, \ln{[f{(x)}}]}$ $\,=\,$ ${\dfrac{1}{f{(x)}}}{\dfrac{d}{dx}{\, f{(x)}}}$

Other forms

The derivative rule for the natural logarithm of a function can be written in terms of any function and variable.

$(1) \,\,\,$ $\dfrac{d}{dm}{\, \log_{e}{g{(m)}}}$ $\,=\,$ ${\dfrac{1}{g{(m)}}}{\dfrac{d}{dm}{\, g{(m)}}}$

$(2) \,\,\,$ $\dfrac{d}{dp}{\, \log_{e}{r{(p)}}}$ $\,=\,$ ${\dfrac{1}{r{(p)}}}{\dfrac{d}{dp}{\, r{(p)}}}$

$(3) \,\,\,$ $\dfrac{d}{dz}{\, \log_{e}{s{(z)}}}$ $\,=\,$ ${\dfrac{1}{s{(z)}}}{\dfrac{d}{dz}{\, s{(z)}}}$

Proof

Learn how to derive the differentiation formula for the natural logarithm of a function.

Problems

List of solved problems to learn how to use the derivative of natural logarithm of a function formula in differential calculus.