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Section 8.1 The Single Variable Chain Rule

Recall that the chain rule for functions of one variable says:

\begin{equation*} \text{If } y=y(x) \text{ and } x=x(t) \text{ then } \dfrac{dy}{dt} = \dfrac{dy}{dx}\cdot \dfrac{dx}{dt}. \end{equation*}

Use the chain rule to find \(\dfrac{df}{dx}\) if \(f(u) = \sin(u)\) and \(u(x)=x^2+1\text{.}\)

Answer.

\(\dfrac{df}{dx} = 2x\cos(x^2+1)\)

Solution.

Via the chain rule:

\begin{alignat*}{1} \dfrac{df}{dx} \amp = \dfrac{df}{du}\dfrac{du}{dx}\\ \quad \amp = \cos(u)(2x)\\ \quad \amp = 2x\cos(x^2+1). \end{alignat*}

With multivariable functions there are many ways in which to form composite functions but there will be a chain rule for each possibility. In the following sections we will look at some of these.