Chapter 4.1, Problem 36E

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a) Let n be a postive integer. In this part a) you will prove the Power Rule for the root functions y 1 Vx = x* on (0, ). Use inverse functions to show that if f(x) x" then the inverse dy function y = f-1(x) = x* has the derivative dx 1r-1. n b) Let n be a postive integer and m be an integer. In this part b) m т you will prove the Power Rule for rational exponents y = xn = on the interval (0, ∞). d Assume the result dx --1 of part a) and use the Chain Rule to show that d m m -1 dx n

5 true: Given f(x) = ar + In(cos()). Find the value(s) of a that make the statement f(-) Question Help: Message instructor Submit Question Jump to Answer

3) Use the information below about the functions fand h to answer the following COS X Let h(x)= (a) Find h'(x) = = (b) Find h'(0) = .f(x) X -2 -1 0 1 2 f(x) 3 -9 5 3 -5 f'(x) -5 4 9 2 8 (c) Based on your answer for (b), determine if x = 0 is a critical number of h(x) or not. Explain why.

= cos-. a State the domain of f(x). (Hint: Think about it rather than relying on algebra.) 17 Consider the function f(x) 1 b Recalling that Vx² : |x], show that f'(x) = |x|Vx? - 1 c Comment on f'(1) and f'(-1). d Use the expression for f'(x) in part b to write down separate expressions for f'(x) when x > 1 and when x 1 and for x < -1. f Find: i lim f(x) ii lim f(x) X- 00 X - 00 g Sketch the graph of y = f(x).

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af af af 5) Find the following of the function f(x,y) =xy+cos y+ysin x

2.(9) Determine the derivative of g Let g(u) = 7 8u+4