Identifier: Derivatives Inside Derivatives

Overall Shape when you squint your eyes.

Primary Derivative Rule

Example

$y=\left(\mathit{ }\right)\left(\mathit{ }\right)$

$y=\left(\mathit{SOMETHING}\right)\mathit{\bullet }\left(\mathit{SOMETHING}\right)$

Something inside a set of parentheses multiplied by something else in a set of parentheses.

Product Rule:

$fg‘+gf‘$

$f\left(x\right)={\mathrm{(}{5x}^3–45x^2+35x\mathrm{)}}^6{\mathrm{(}5x^4+3x^2\mathrm{)}}^7$

Primarily this is two chunks being multiplied.

$y=\frac{\left(\mathit{ }\right)}{\left(\mathit{ }\right)}$

$y=\frac{\left(\mathit{SOMETHING}\right)}{\left(\mathit{SOMETHING}\right)}$

Something divided by Something.

Quotient Rule:   $\frac{\left(\mathit{low}\right)\left(\mathit{dhi}\right)\mathit{–}\left(\mathit{hi}\right)\left(\mathit{dlow}\right)}{{\left(\mathit{low}\right)}^2}$

$f\left(x\right)=\frac{{({3x}^3–6x+9)}^4}{{(5x^4+3x^2)}^3}$

Primarily this is two chunks being divided.

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

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

Something inside of something.

A function inside another function.

Chain Rule:

( DIN )( DOUT )

${y=\mathrm{(}\frac{{3x}^3–6x+9}{5x^4+3x^2}\mathrm{)}}^5$

Primarily this is an equation inside another equation.

Special cases

You have to have the special case options memorized to be able to identify them.

Like we talked about when we were learning all of these special case derivatives, once it is more than just a basic x inside the, it is a chain rule.

$f\left(x\right)=\mathit{ln}\left(\mathit{sin}\right(x\left)\right)$

In this example the primary rule is the chain rule. The natural log special case acting as the outside has the sine function inside it.