Chain Rule

Formula

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

A formula that calculates the derivative of the composition of the functions is called the chain rule.

Introduction

Two or more functions can form a function by their composition to represent a quantity in mathematics. The derivatives of such composite functions cannot be calculated directly by the differentiation rules due to complexity in the formation of the functions.

However, the differentiations of composite functions can be evaluated by the same derivative properties with a special law and it is called the chain rule. It is expressed mathematically in calculus in the following two forms.

Basic form

Let $f(x)$ and $g(x)$ be two functions. If their composition forms a special composite function $f\Big(g(x)\Big)$, then the chain rule is written as follows.

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

Learn how to derive the chain rule in fundamental notation in differential calculus.

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Leibniz’s notation

If $y \,=\, f\Big(g(x)\Big)$ and $z \,=\, g(x)$, then the chain rule is written in the following Leibnitz’s form.

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

Learn how to prove the chain rule in Leibnitz’s form mathematically in differential calculus.

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Example

Find $\dfrac{d}{dx}\,\cos{\Big(\log_{e}{(x)}\Big)}$

There are differentiation properties for finding the derivatives of cosine and logarithmic functions but no rule to evaluate the derivative of cosine of natural logarithm of $x$ with respect to $x$ and it is not possible to find its derivative by those rules directly due to the composition of the functions. However, the same laws can be used to find derivative of any composite function by the chain rule.

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