Chain rule in Leibniz’s notation

Formula

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

Introduction

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.

Example

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.

Proof

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

Problems

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