Chain rule in fundamental notation

Formula

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

Introduction

Let $f(x)$ and $g(x)$ be two functions in $x$ and their composition forms a special function $f\Big(g(x)\Big)$. There are many differential properties to find the derivatives of the functions but they cannot be used as formulas to find the derivative of any composite function due to the involvement of two or more different functions.

However, the same differential laws can be used to find the differentiation for the composition of two or more functions by a special rule, and it is called the chain rule.

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

The chain rule is written fundamentally in the above mathematical form and it states that the derivative of a composite function is equal to the product of the derivative of the function with respect to its internal function and the derivative of the internal function with respect to its variable.

Example

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

In this problem, $e$ raised to the power of $\sin{x}$ is a function and it is formed by the composite of an exponential function and a sine function.

Proof

Learn how to prove the chain rule in fundamental notation to find derivative of any composition function.

Problems

List of the problems on chain rule to learn how to use the fundamental notation for finding the derivative of composition of two or more functions.

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