Question

In the text we noted that if $f\left(u\right(v\left(x\right)\left)\right)$ was a composition of three functions, then its derivative is $\frac{df}{dx}=\frac{df}{du}\times \frac{du}{dv}\times \frac{dv}{dx}$. Write this rule in “prime” notation.

Short answer

In prime notation the derivative is:

$\left(f\circ u\circ v\right)\text{'}\left(x\right)=f\text{'}(u\circ v)\left(x\right)\times (u\circ v)\text{'}\left(x\right)\times v\text{'}\left(x\right)$

Step-by-step solution

Step 1. Given information:

The composite function and its derivative are:

$f\left(u\right(v\left(x\right)\left)\right)$

$\frac{df}{dx}=\frac{df}{du}\times \frac{du}{dv}\times \frac{dv}{dx}$

Step 2. Derivative in the prime notation:

The derivative of the given composite function is written as:

$\left(f\circ u\circ v\right)\text{'}\left(x\right)=f\text{'}(u\circ v)\left(x\right)\times (u\circ v)\text{'}\left(x\right)\times v\text{'}\left(x\right)$