Math 165 – Quiz 5D, inverse trig functions – solutions Problem 1 Consider the function f (x) = tan−1 (0.5x). a) Find f 0 (x). b) Which of the graphs below is the graph of f (x)? Circle A, B, or C, or write ’none’. A B C Solution a) Use the Constant Multiple Rule and the Chain Rule with u = 0.5x and du/dx = 0.5 to get f 0 (x) = 1 0.5 = . 2 1 + (0.5x) 4 + x2 b) The correct answer is A. You can graph f (x) to see this, here is what you can see without a calculator. • tan−1 (u) is defined for all u, so f (x) is defined for all x, never has a vertical tangent (rules out B, C). • f 0 (x) > 0 for all x, so the graph of f (x) must be going up (f (x) is increasing, rules out C). • tan−1 (u) is an odd function, same as f (x) (rules out B).