The Derivatives of Composite FunctionsThe Derivatives of Composite Functions "I can find the derivative function using the Chain Rule in function prime notation, and Leibniz notation.

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.