# 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

## Recently Viewed Presentations

• Presentation for Droga Gunnar Hilsen Product Manager by Marin Sobin Izola, Slovenia 12.12.2006.
• Art. 207 TFEU. Object of EU trade policy "The common commercial policy shall be based on uniform principles, particularly with regard to changes in tariff rates, the conclusion of tariff and trade agreements relating to trade in goods and services,...
• Cavity QED (Mostly) coherent dynamics. Infinite-range interactions. All atoms between the cavity mirrors interact. Range not tunable. Infinite-range interactions are a "weird" case. The high symmetry often reduces the complexity of the system.
• Let's Practice. With your partner, go to an open space and practice these 4 stretches. One will be a judge, the other the competitor. Don't forget to do the warm up!!!! Make sure you study which muscle group involved and...
• Chronic venous insufficiency Management Chronic venous insufficiency Management Chronic venous insufficiency * Chronic venous insufficiency (C.V.I.) encompasses disease of the lower limb veins in which venous return is impaired usually over a number of years of reflux, obstruction or calf...
• Report from the GDE director Barry Barish Snowmass 14-Aug-05
• A Biome has a characteristic climax community The earth has two main types of biomes, land biomes and aquatic biomes. Most land biomes are named for their climax community or dominant type of plant life. The major types of biomes...
• The output of a delta modulator is a bit stream of samples at a relatively high rate, the value of each bit being determined according to whether the input message sample amplitude has increased or decreased relative to the previous...