Since y = tan ^{1}x is the inverse of the function y = tan x, the function y = tan ^{1}x if and only if tan y = x. But, since y = tan x is not onetoone, its domain must be restricted in order that y = tan ^{1}x is a function. To get the graph of y = tan ^{1}x, start with a graph of y = tan x. (The window at right is [2 , 2] x [4, 4]. ) 

Restrict the domain of the function to a onetoone region  typically is used (highlighted at right) for tan ^{1}x. This leaves the range of the restricted function unchanged as .  
Reflect this graph across the line y = x to get the graph of y = tan ^{1}x (y = arctan x), the thickest black curve at right. Notice that y = tan ^{1}x has domain and range . It is strictly increasing on its entire domain. 

So, when you ask your calculator to graph y = tan ^{1}x, you get the graph shown at right. (The viewing window is [2 , 2] x [4, 4].) 
The derivative of y = tan ^{1} x is: (Click here for a derivation.)
The graphs of y = tan ^{1} x and its derivative is shown below. The domain of y' is . Since y = tan ^{1} x is always increasing, y' > 0 for all x in its domain.
Here is a typical problem:
What is the area under the graph of on [0,10]?
The graph of this region is shown at right. The area of this region is:
Now, we know that:
Comparing what we know with what we need to know suggests that we let u^{2} = 4x^{2}: