Math Doubts

Chain rule in Leibniz’s notation


$\dfrac{dy}{dx} \,=\, \dfrac{dy}{dz} . \dfrac{dz}{dx}$


Chain rule is a fundamental method in differential calculus to find the derivative of the composition of functions. The chain rule is expressed in standard notation as follows.

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

It creates some inconvenience while writing it every time. So, it is always recommended to write it in simple and easier form. For this reason, assume $y \,=\, f\Big(g(x)\Big)$ and $z \,=\, g(x)$. Now, substitute them in the above differential equation.

$\therefore\,\,\,\,\,\,$ $\dfrac{dy}{dx} \,=\, \dfrac{dy}{dz} \times \dfrac{dz}{dx}$

It is called the chain rule in Leibniz’s notation.


Find $\dfrac{d}{dx}\,e^{\displaystyle x^2}$

It is a composite function, which is formed by the composition of an exponential function and a power function. So, it is not possible to find its derivative by either of them. However, it is possible to calculate its differentiation by using chain rule.

Assume $y \,=\, e^{\displaystyle x^2}$ and $z \,=\, x^2$. Now, let’s find the derivative of the composite function by substituting them in the chain rule in Leibnitz’s form.


Learn how to derive the chain rule in Leibnitz’s form to find the derivative of the composition of the two or more functions.


List of the questions on finding the derivatives of composite functions by chain rule in Leibniz’s notation.

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 - 2021 Math Doubts, All Rights Reserved