Question

Use the chain rule twice to prove that$\frac{d}{dx}\left(f\left(u\left(v\left(x\right)\right)\right)\right)=f\text{'}\left(u\left(v\left(x\right)\right)\right)u\text{'}\left(v\left(x\right)\right)v\text{'}\left(x\right)$

Short answer

$$\begin{gathered} \frac{d}{dx}\left(f\left(u\left(v\left(x\right)\right)\right)\right) \\ =\frac{d}{dx}f\left(u\left(v\left(x\right)\right)\right)\times \frac{d}{dx}u\left(v\left(x\right)\right) \\ =f\text{'}\left(u\left(v\left(x\right)\right)\right)\frac{d}{dx}u\left(v\left(x\right)\right) \\ =f\text{'}\left(u\left(v\left(x\right)\right)\right)u\text{'}\left(v\left(x\right)\right)\frac{d}{dx}v\left(x\right) \\ \Rightarrow \frac{d}{dx}u\left(v\left(x\right)\right)=f\text{'}\left(u\left(v\left(x\right)\right)\right)u\text{'}\left(v\left(x\right)\right)v\text{'}\left(x\right) \end{gathered}$$

Hence proved.

Step-by-step solution

Step 1. given Information

We have to prove the following derivative :-

$\frac{d}{dx}\left(f\left(u\left(v\left(x\right)\right)\right)\right)=f\text{'}\left(u\left(v\left(x\right)\right)\right)u\text{'}\left(v\left(x\right)\right)v\text{'}\left(x\right)$

We have to use chain rule twice to prove this derivative.

Step 2. Proof of derivative

Take the left hand side :-

$\frac{d}{dx}\left(f\left(u\left(v\left(x\right)\right)\right)\right)$

Use chain rule on the function$f$ :-

role="math" localid="1648908948871" $$\begin{gathered} \frac{d}{dx}\left(f\left(u\left(v\left(x\right)\right)\right)\right)=\frac{d}{dx}f\left(u\left(v\left(x\right)\right)\right)\times \frac{d}{dx}u\left(v\left(x\right)\right) \\ \Rightarrow \frac{d}{dx}\left(f\left(u\left(v\left(x\right)\right)\right)\right)=f\text{'}\left(u\left(v\left(x\right)\right)\right)\frac{d}{dx}u\left(v\left(x\right)\right) \end{gathered}$$

Now use chain rule again on the function $u$, then we have :-

role="math" localid="1648908979607" $$\begin{gathered} \frac{d}{dx}u\left(v\left(x\right)\right)=f\text{'}\left(u\left(v\left(x\right)\right)\right)u\text{'}\left(v\left(x\right)\right)\frac{d}{dx}v\left(x\right) \\ \Rightarrow \frac{d}{dx}u\left(v\left(x\right)\right)=f\text{'}\left(u\left(v\left(x\right)\right)\right)u\text{'}\left(v\left(x\right)\right)v\text{'}\left(x\right) \end{gathered}$$

This is equal to right hand side.

Hence proved.