Identifier: Chain Rule Derivative

The chain rule is used on what are called composite functions, I like to say functions inside functions.

If you look down at your problem and the primary operation you see is functions inside the parenthesis of other functions, then the chain rule is what you will want to use.

I like to think about these in terms of shapes, SOMETHING inside SOMETHING . So, functions made up of multiple functions. These composite functions can have many different looks to them, but the main setup often looks like this:

$f\left(x\right)={\left(\mathit{SOMETHING}\right)}^{\mathit{Something}}$

$f\left(x\right)={\left(\mathit{IN}\right)}^{\mathit{OUT}}$

$f\left(x\right)={(5x^4+3x^2)}^{\frac{3}{4}}$

An equation inside a set of parentheses raised to a power.

You will see that all of the derivatives we will cover in the Special Case section are very often chain rules once they become more than their basic cases.