Math Doubts

Proof of Chain Rule


$\dfrac{d}{dx} {f[{g(x)}]} \,=\, {f'[{g(x)}]}.{g'{(x)}}$

Let $f(x)$ and $g(x)$ be two functions in terms of $x$ and their composition formed a composite function, denoted by $f\Big(g(x)\Big)$. The derivative of $f\Big(g(x)\Big)$ with respect to $x$ is written mathematically as follows in calculus.



$f{(x)}$ is a function and $g{(x)}$ is another function. $f{(g{(x)})}$ is a composition of the both functions.

Representation of Derivatives of the functions

The differentiation of a function is represented in short form in calculus as follows. It is used in deriving the chain rule. So, remember them.

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

$(2) \,\,\,\,\,\,$ $\dfrac{d}{dx} g{(x)} \,=\, g'{(x)}$

The derivative of function $f{[g{(x)}]}$ with respect to $x$ is written mathematically as follows.


Transform the function in known form

Generally, $f{(x)}$ is a known form of the function. So, try to convert the function $f{[g{(x)}]}$ to the known form for simplifying the complexity of the function.

Take $y = g{(x)}$. Therefore, the function $f{[g{(x)}]}$ can be simplify written as $f{(y)}$.

$\implies \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $\dfrac{d}{dx}{f{(y)}}$

Change the differential element

The function $f{(y)}$ is in terms of $y$ but the differential element is in terms of $x$. It is not possible to differentiate the function $f{(y)}$ with respect to $x$. Therefore, convert the differential element ($dx$) from $x$ to $y$.

It is possible by differentiating $y = g{(x)}$ with respect to $x$.

$\dfrac{d}{dx}{y} \,=\, \dfrac{d}{dx}{g{(x)}}$

$\implies \dfrac{dy}{dx} \,=\, g'{(x)}$

$\implies \dfrac{dy}{g'{(x)}} \,=\, dx$

$\,\,\, \therefore \,\,\,\,\,\, dx \,=\, \dfrac{dy}{g'{(x)}}$

Now, replace the differential element from $x$ to $y$ term.

$\implies \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $\dfrac{d}{\dfrac{dy}{g'{(x)}}}{f{(y)}}$

The function $g'{(x)}$ divides the differential element $dy$. So, it multiplies $d$ in numerator.

$\implies \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $\dfrac{g'{(x)} \times d}{dy}{f{(y)}}$

$\implies \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $g'{(x)} \dfrac{d}{dy}{f{(y)}}$

$\implies \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $g'{(x)} f'{(y)}$

Get the Chain Rule

Now, replace the value of $y$. Actually, it is $y = g{(x)}$. Therefore, replace $y$ by its value in the composition of the functions.

$\implies \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $g'{(x)} f'{[{g{(x)}}]}$

$\,\,\, \therefore \,\,\,\,\,\, \dfrac{d}{dx}{f{[g{(x)}]}}$ $\,=\,$ $f'{[{g{(x)}}]}g'{(x)}$

This property is called as chain rule in differential calculus and it is used as a formula while dealing the functions which are formed compositions of two or more functions.

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the maths problems in different methods with understandable steps.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved