# 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.

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Jun 26, 2023

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