Derivatives Practice Math 210 211 Power Rule If f x ax n then f x anx n 1 dx Note this rule applies to radicals fractional exponents and simple rational expressions negative exponents The dx that trails is meant to signify the chain rule If x the base happens to be a function itself then you must apply the chain rule Technically you always apply the chain rule but if the base is simply x then the chain rule gives 1 as the derivative and the answer is unchanged If f x 4 x 3 then f x 12 x 2 1 The chain rule gives a 1 since it s the derivative of x but we usually just write the answer as f x 12 x 2 Example Example If f x 3 x 2 2 5 then f x 15 x 2 2 4 2 x In this case the chain rule gives a derivative of 2x which trails the answer We would normally clean up the answer and write f x 30 x x 2 2 4 Exponential Rule Example If f x e x then f x e x dx f x e 2 x f x 2e 2 x Natural Logarithm Rule Example If f x ln x then f x f x ln 2 x 3 x 1 f x 1 dx dx x x 1 6x 2 1 2 6 x 1 2x3 x 1 2x 3 x 1 Note how the final answer is combined into one expression for convenience Product Rule If f x u v then f x u v v u Quotient Rule If f x u v u u v then f x v v2 Practice Answers 1 f x x 2 3x 1 2 f x 4 x 6 1 5 x 4 3 x 3 x 17 3 f x 2 x 2 x 2 4 f x 3 x 2 5 12 13 f x 3 x 2 1 3 g x 3 x 4 g x 2 x 1 2 5 g x 4 2x 2 h x e 3 x x 2 h x e 2 x 1 4 j x ln x x 2 2 x 3 j x ln e x 2 j x x ln x 1 14 k x e ln x 15 k x ln x 1 x 3 16 k x 6 7 8 9 10 11 17 3x e5x ln 2 x x2 1 m x 3 x 1 2 19 e 2 x ln x m x x2 n x 25 2 e 20 p x 18 x ex In no particular order Some have been simplified 4x 4 6x 2 2x x 3 1 3 8e 2 x e 2 x 1 3 x ln x 1 x 1 10 5 x 2x 3 16 x 2 x 1 3 ex ex 2 2 6 x 1 e 3 x x 2 e3x 3e 3 x ln x x 9 4 x 1 2 x 6x 2 x x 2 2x3 1 ex 1 x 2 x ex x 3 x2 8 x 3 12 x 2 4 x 0 24 x 5 6 x 3 9 x 2 1 2x 33 1 x 2 2 3 x 2 ln x 1 x 2 x 1 5x 5 xe ln 2 x e 5 x x ln 2 x 2 Any errors real or imagined contact surgent asu edu Good luck Updated 8 26 09
View Full Document
Unlocking...