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

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.

$\dfrac{d}{dx}\,f\Big(g(x)\Big)$

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

$\dfrac{d}{dx}{f{[g{(x)}]}}$

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 $f{(y)}$.

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

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.

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

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

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

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.

Latest Math Topics

Apr 18, 2022

Apr 14, 2022

Apr 05, 2022

Mar 18, 2022

Mar 05, 2022

Latest Math Problems

Apr 06, 2022

Mar 22, 2022

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

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

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

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved