Math Doubts

Proof of Chain Rule

Formula

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

Proof

$f{(x)}$ 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.

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

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.



Follow us
Email subscription
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
Mobile App for Android users Math Doubts Android App
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