The Chain Rule Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002 U.S.S. Alabama Mobile, Alabama online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Chain Rule: If is the composite of and, then: online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Here is the chain rule in action: online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Here is the chain rule in action: online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Here is the chain rule in action: online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Here is the chain rule in action: online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Here is the chain rule in action: online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Here is the chain rule in action: Differentiate the outside function... …then the inside function online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Another example: Find the derivative of

Another example: Find the derivative of It is helpful to identify the outside function and the inside function. In this example, the outside function is the cube, and the inside function is x

Another example: Find the derivative of It is helpful to identify the outside function and the inside function. In this example, the outside function is the cube, and the inside function is x The chain rule says take the derivative of the outside function leaving the inside function unchanged and then multiply by the derivative of the inside function.

Another example: Find the derivative of It is helpful to identify the outside function and the inside function. In this example, the outside function is the cube, and the inside function is x The chain rule says take the derivative of the outside function leaving the inside function unchanged and then multiply by the derivative of the inside function. The derivative of the inside using the Power Rule The derivative of the outside leaving the inside unchanged

Another example: Find the derivative of It is helpful to identify the outside function and the inside function. In this example, the outside function is the cube, and the inside function is x The chain rule says take the derivative of the outside function leaving the inside function unchanged and then multiply by the derivative of the inside function. The derivative of the inside using the Power Rule The derivative of the outside leaving the inside unchanged Finally we need to simplify the answer.

Look at this one: Differentiate the outside function... …then the inside function online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc03_6.ppt

Example: Find the derivative of

Example: Find the derivative of The outside function is the cube root function and the inside function is. First rewrite the function with rational exponent:

Example: Find the derivative of The outside function is the cube root function and the inside function is. First rewrite the function with rational exponent: To find the derivative of the outside, do the Power Rule: Note: This is just HOW we are finding the derivative of the outside function.

Now do a little simplification: Multiply the 1/3 and the 6x. Now let’s look at the actual derivative using the Chain Rule. The derivative of the outside leaving the inside unchanged The derivative of the inside

Example: Find the derivative of:

Example: Find the derivative of: Rewrite and use the chain rule.

Example: Find the derivative of: Rewrite and use the chain rule. Don’t mess with the inside!!! derivative of the inside

Example: Find the derivative of: Rewrite and use the chain rule. Don’t mess with the inside!!! derivative of the inside The result needs some simplifying: Multiply the constants together.