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Math_137_Winter_2010_Week_5_Notes

Math_137_Winter_2010_Week_5_Notes - Math 137 Week 5 Notes...

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Math 137 Week 5 Notes – Dr. Paula Smith 3.1 The derivative of a constant (function) is zero. Power Rule : Given a power function f(x) = x n , when n is a positive integer, 1 2 2 3 3 2 2 1 ... + + + + + + = n n n n n n n n x y x y x y x xy y x y x y (because ) )( ... ( 1 2 2 3 3 2 2 1 x y x y x y x y x xy y n n n n n n + + + + + + n n n n n n n n n n n n x y x y x y x y x xy y x y x y x y x xy y + + + + + + = 1 2 2 3 3 2 2 1 1 2 2 3 3 2 2 1 ... ... n n x y = ), so 1 1 2 2 3 3 2 2 1 ... lim lim = + + + + + + = n n n n n n n x y n n x y nx x y x y x y x xy y x y x y . Hence f (x) = nx n--1 . In fact, this is true for all real numbers n, including n = ½ , n = – 1, etc. If c is a constant multiple of function f(x), then ) ( ) ( ) ( lim ) ( ) ( lim ) ( ) ( lim ) ) ( ( 0 0 0 x f c h x f h x f c h x f h x f c h x cf h x cf x cf h h h = + = + = + = If f and g are both differentiable functions, then x y x g y g x f y f x y x g x f y g y f x g f x y x y
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