The Derivatives of Composite Functions

"I can find the derivative function using the Chain Rule in function prime notation, and Leibniz notation. I can apply it in various contexts."

Function Composition

Example Given:

Find: a) b) c)

A countless number of functions, including parent functions studied, can be expressed as composite functions. Hence a derivative function for composite functions is useful.

Chain Rule

Now prove it...

Example If find in simplified/factored form.

("prime" notation)

Chain Rule("Leibniz" notation)

Example If and , find at

Example Differentiate:

Express in simplified/factored form.

(Can easily be proven)

Power of a Function Rule (an encore presentation)

Could we have differentiated the previous function after a previous lesson?

Yes! The Power of a Function Rule is a particular case of The Chain Rule:

A lot of functions....Page 105...#1f, 4e, 5ad, 6, 8c, 9a, 10, 13ad, 14, 15, 18

A3. I can verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems;

Example Given: Find: a) b) c)

Example If find in simplified/factored form.

Example If and , find at

Example Differentiate:

Express in simplified/factored form.