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Section 4.5 Chain rule

We now study another differentiation rule, known as the “chain rule”. It is useful to calculate the derivative of a “composite function”, namely a “function of a function”. For instance, if \(f\) and \(g\) are differentiable functions, then the chain rule can be used to calculate the derivative of the composite function \(F(x) = f(g(x))\text{.}\)

Subsection 4.5.1 Instructional video

Subsection 4.5.2 Key concepts

Concept 4.5.1. The chain rule.

Let \(f\) and \(g\) be two functions such that \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g(x)\text{.}\) Then the composite function \(F = f \circ g \) defined by \(F(x) = f(g(x))\) is differentiable at \(x\text{,}\) and its derivative is given by

\begin{equation*} F'(x) = f'(g(x)) \cdot g'(x). \end{equation*}

This is known as the chain rule.

In Leibniz notation, if \(y = f(u)\) and \(u = g(x)\text{,}\) then the chain rule can be written as

\begin{equation*} \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}. \end{equation*}

A tip: When using the chain rule, you work from the outside to the inside. You first differentiate the outer function \(f\) (evaluated at the inner function \(g(x)\) ) and then multiply by the derivative of the inner function.

Concept 4.5.2. Do we need the quotient rule?

We derived previously the quotient rule. But do we really need it? Instead of using the quotient rule, you can always use the product rule and the chain rule instead. Indeed,

\begin{equation*} \frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{d}{dx} \left( f(x) (g(x))^{-1} \right), \end{equation*}

and then you can calculate the right-hand-side by using the product rule and the chain rule.

Further readings 4.5.3 Further readings