Chain Rule

105. Chain Rule

Compute the derivative of a composite function

\begin{aligned} ℎ^{'} (𝑥) & = \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑔 (𝑥))] \\ = 𝑓^{'} (𝑔 (𝑥)) \cdot 𝑔^{'} (𝑥) \end{aligned}

More generally:

𝑦 = 𝑓_{1} (𝑓_{2} (𝑓_{3} (\dots 𝑓_{𝑛} (𝑥) \dots)))

\frac{𝑑 𝑦}{𝑑 𝑥} = 𝑓_{1}^{'} (𝑓_{2} (𝑓_{3} (\dots 𝑓_{𝑛} (𝑥) \dots))) \cdot 𝑓_{2}^{'} (𝑓_{3} (\dots 𝑓_{𝑛} (𝑥) \dots)) \cdot 𝑓_{3}^{'} (\dots 𝑓_{𝑛} (𝑥) \dots) \cdot \dots \cdot 𝑓_{𝑛}^{'} (𝑥)

Example

\frac{𝑑}{𝑑 𝑥} [𝑓 (𝑔 (𝑥))] = 𝑓^{'} (𝑔 (𝑥)) \cdot 𝑔^{'} (𝑥)

\frac{𝑑}{𝑑 𝑥} [𝑒^{sin (𝑥)}]

\frac{𝑑}{𝑑 𝑥} [𝑒^{𝑥}] = \frac{1}{𝑥} \frac{𝑑}{𝑑 𝑥} [sin (𝑥)] = cos (𝑥)

𝑓^{'} (𝑔 (𝑥)) = \frac{1}{sin (𝑥)} 𝑔^{'} (𝑥) = cos (𝑥)

\frac{𝑑}{𝑑 𝑥} [\underset{𝑓 (𝑔 (𝑥))}{\underset{⏟}{ln (\overset{𝑔 (𝑥)}{\overset{⏞}{sin (𝑥)}})}}] = \frac{1}{sin (𝑥)} \cdot cos (𝑥)

𝑓 (𝑥) = {cos}^{3} (𝑥) = {(cos (𝑥))}^{3}

𝑓 (𝑥) = 𝑣 (𝑢 (𝑥))

𝑓^{'} (𝑥) = 𝑣^{'} (𝑢 (𝑥)) \cdot 𝑢^{'} (𝑥)

\begin{aligned} 𝑓^{'} (𝑥) & = \frac{𝑑 𝑣}{𝑑 𝑢} \cdot \frac{𝑑 𝑢}{𝑑 𝑥} \\ = \frac{𝑑 {(cos (𝑥))}^{3}}{𝑑 cos (𝑥)} \cdot \frac{𝑑 cos (𝑥)}{𝑥} \\ = 3 {(cos (𝑥))}^{2} \cdot - sin (𝑥) \\ = - 3 {(cos (𝑥))}^{2} sin (𝑥) \end{aligned}