Math Doubts

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.



Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more