# 3.1 Shortcut Derivative Rules - University of Houston 3.2 The Product and Quotient Rules The Product Rule [ f ( x) g ( x)]' f ' ( x) g ( x) f ( x) g ' ( x) The Quotient Rule f ( x) f ' ( x) g ( x) f ( x) g ' ( x) [ ]' g ( x) [ g ( x)]2 1 3.2 Using the Product & Quotient Rules

Find the derivative of each function. 1. F(x)=(3x2+4) (x3-2) 2. 3. 4. f ( x) x 4 ( x 2) x3 1 h( x ) 2x 5 x3/ 2 h( x ) 4 2x 1 2 3.2 Application Example Worldwide sales, in millions of dollars, of a DVD

recording of a hit movie t months from the date of release is given by 7t S (t ) 2 t 1 Find the rate at which sales are changing at time t. How fast are sales changing: At the time the DVD is released (t=0)? Six months from the date of release? Answer in complete sentences. 3 3.3 The Chain Rule Algebra review. In composition of functions: the output of one function is used as input to another

function; there is an inside function and an outside function. If f(x)=x2 and g(x)=5x+1, then ( f g )( x) f ( g ( x)) f (5 x 1) (5 x 1) 2 and ( g f )( x) g ( f ( x)) g ( x 2 ) 5 x 2 1 Note the answer to the first example above is a function raised to a power. Whenever you have a function raised to a power, like this example, you can think of composition and a function inside of a function. 4 3.3 The Chain Rule If h(x)=g[f(x)], so g is the outside function and f is the inside function, then the derivative of h equals the derivative of the outside function g at the

inside function f times the derivative of the inside function f. In symbols, h(x)=g[f(x)] f (x) General Power Rule (special case of chain rule) {[f(x)]n}=n[f(x)]n-1 f (x) 5 3.3 Using the Chain Rule/General Power Rule Find the derivative of each function. 1. f(x)=(3x+1)5 2. g ( x) 3x 4 5 3.

3 h(t ) 2 (7t 1) 6 4. 3x 6 f ( x) ( 2 ) 4 x 1 6