# What is the Chain Rule for derivatives?

Apr 22, 2018

$f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

#### Explanation:

In differential calculus, we use the Chain Rule when we have a composite function. It states:

The derivative will be equal to the derivative of the outside function with respect to the inside, times the derivative of the inside function. Let's see what that looks like mathematically:

Chain Rule:

$f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

Let's say we have the composite function $\sin \left(5 x\right)$. We know:

$f \left(x\right) = \sin x \implies f ' \left(x\right) = \cos x$

$g \left(x\right) = 5 x \implies g ' \left(x\right) = 5$

So the derivative will be equal to

$\cos \left(5 x\right) \cdot 5$

$= 5 \cos \left(5 x\right)$

We just have to find our two functions, find their derivatives and input into the Chain Rule expression.

Hope this helps!