**The Chain Rule**

Why do we use the chain rule?

The chain rule is an effective and easier way of finding the derivative of complex equations.

The chain rule: it is a useful tool!

How to use the Chain Rule

In using the chain rule, you need to work from the outside to the inside. First, differentiate the outer function and then multiply by the derivative of the inner function(s).

BE CAREFUL! It can get messy, so pay attention to detail, don't rush through the math, and don't get cocky (we speak from experience)!

**By: Mauli Saini, Hannah Thornton, & Josh Ramnath**

How do I know when to use the chain rule?

ANYTIME you see a function "nested" inside of another function, think to use the chain rule.

Aside from saving you a lot of time and calculations, it also lets you differentiate functions that may seem impossible to derive.

For example, if we try to multiply out f(x) = (3x^2 + 4x - 5)^2 and then differentiate it. Slightly manageable, right? Now, imagine if we try to multiply out f(x) = (3x^2 + 4x - 5)^5. That would not be worth the time, but with the chain rule it becomes very easy!

The Chain Rule in Action:

[

f(

g(x)

)

]'

= f'(

g(x)

) *

g'(x)

f(x) =

(

3x^2 + 4x - 5

)^2

f'(x) =

2

(

3x^2 + 4x - 5

) *

(6x + 4)

Animation time!

http://goanimate.com/go/movie/0tM4BBJEUQpE?utm_source=emailshare&uid=0hbntPabwh00

Differentiate: f(x)=(3x+1026)^2

f '(x)= 2 (3x+1026)(3)

f '(x)= 6 (3x+1026)

And now you know the chain rule!

f(x) = x^3[2x-1]^2

f '(x)= 3x^2[2x-1]^2 + x^3 (2) (2x-1)^1 (2)

f '(x)= 3x^2[2x-1]^2+ 4 x^3 (2x-1)^1

y= [sin(5x)]

y'=[cos(5x)](5)

y'= 5[cos(5x)]

Example #2

Example #3

Example #4

The derivative of sine

The derivative of (5x)

Just bring down the function given in parenthesis

Clean it up!

The derivative of (3x+1026)

The function in parenthesis

Take the 2 from the exponent and subtract that by 1.

Clean it up