Unformatted text preview: k . 5. a) Prove that the formula for the derivative of an inverse function is ( f − 1 ) ( x ) = 1 f ( f − 1 ( x )) . (Hint: Let g ( x ) = f − 1 ( x ), then f ( g ( x )) = x . DiFerentiate.) b) What is (arccos( x )) ? Write an expression which is free of trig functions. (Hint: Use the formula in part a . To help you simplify, draw a triangle.) c) Let y = tan 2 (arccos( x )). ±ind dy dx using the chain rule. d) ±ind another expression for y which doesn’t mention trig functions. (Hint: Again, draw a triangle.) e) ±ind dy dx from your expression in part d). How does it compare with your answer in part a)? 6. What is f (4) if f (4) = 2 and x 2 f ( x ) = 6 x + ( f ( x )) 3 for x near 4? 7. A diFerentiable function y ( x ) satis²es x 2 cos y + sin y = x and y (1) = 0. What is y (1)?...
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- Spring '08
- Derivative, Inverse function, dy dx