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).