Chain Rule

Power Rule Product Rule Quotient Rule

We are not able to take the derivative of these functions with the power, product and quotient rules. We will be able to take the derivative of these functions using the Chain Rule To use the Chain Rule the function needs to be broken up into two functions. The next slides will demonstrate how to break up a function to use the chain rule.

I will divide these steps into two parts If I plug in a number for x what happens to it? Multiplied by 2 Add 3 Squared I will divide these steps into two parts

The part of f(x) that does the first two steps is Multiplied by 2 Add 3 Square

The part of f(x) that does the third step is Multiplied by 2 Add 3 Square

This is the inside function Multiplied by 2 Add 3 Square This is the outside function

What is the derivative of the inside function? This is the inside function What is the derivative of the inside function? What is the derivative of the outside function? This is the outside function

Inside= d(Inside)= Outside= d(Outside)= Find the inside function and outside function then find the derivative of each Inside= d(Inside)= Outside= d(Outside)=

Inside= d(Inside)= Outside= d(Outside)= Find the inside function and outside function then find the derivative of each Inside= d(Inside)= Outside= d(Outside)=

Inside= d(Inside)= Outside= d(Outside)= Find the inside function and outside function then find the derivative of each Inside= d(Inside)= Outside= d(Outside)=

The chain rule is This means The Derivative of a Composite Function is: The derivative of the outside function leaving the inside function alone, times the derivative of the inside function

The derivative of the outside function leaving the inside function alone times the derivative of the inside function Derivative of inside function Inside= d(Inside)= Outside= Derivative of outside function leaving inside function alone d(Outside)=

Inside= d(Inside)= Outside= d(Outside)= Derivative of inside function Inside= Derivative of outside function leaving inside function alone d(Inside)= Outside= d(Outside)=

Lets try some on our own.

P146 1-5,7,8

Inside= d(Inside)= Outside= d(Outside)= Derivative of inside function Inside= d(Inside)= Derivative of inside function leaving inside function alone Outside= d(Outside)=

When the last thing to happen is multiplication we need to use the product rule Use the chain rule to find the derivative of the 2nd 1st 2nd = d(1st )=3 d(2nd )

When the last thing to happen is division we need to use the quotient rule Lo= d(Hi)= d(Lo)=

Lets try some on our own.

P146 12,13,15,16

To find the derivative of the inside function I need to use the chain rule a second time.

P146 9-11,14,17-20