Question

Use the product rule to find and state the rule for differentiation of a product of three functions f, g, h in other words$\left(f\right(x\left)g\right(x\left)h\right(x\left)\right)\text{'}$

Short answer

The derivative can be given as

$\left(f\right(x\left)g\right(x\left)h\right(x\left)\right)\text{'}=\left(f\right(x\left)g\right(x\left)\right)h\text{'}\left(x\right)+f\text{'}\left(x\right)g\left(x\right)h\left(x\right)+f\left(x\right)g\text{'}\left(x\right)h\left(x\right)$

Step-by-step solution

Given information

We are given a product of three functions$\left(f\right(x\left)g\right(x\left)h\right(x\left)\right)\text{'}$

Use product rule to find the derivative

We get,

$$\begin{gathered} \left(f\right(x\left)g\right(x\left)h\right(x\left)\right)\text{'} \\ =\left(f\right(x\left)g\right(x\left)\right)h\text{'}\left(x\right)+\left(f\right(x\left)g\right(x\left)\right)\text{'}h\left(x\right) \\ =\left(f\right(x\left)g\right(x\left)\right)h\text{'}\left(x\right)+f\text{'}\left(x\right)g\left(x\right)h\left(x\right)+f\left(x\right)g\text{'}\left(x\right)h\left(x\right) \end{gathered}$$