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

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

Dec 13, 2023

Jul 20, 2023

Jun 26, 2023

Latest Math Problems

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved