Inverse Trigonometric Functions

What we want to do here is use the previous result on the derivative of the inverse function to produce the derivative of an inverse trigonometric function.  For our example, let us take the arcsine function.  Recall that the .

What we must do now is write this in a form that we can evaluate.  To help with that, recall that the range of the arcsine function is .  Note that throughout this interval, the cosine function is nonnegative.

Now our derivative can be evaluated:

The other inverse trigonometric functions can have their derivatives found in the same manner.  Notice that in each case it will be necessary to write a given trigonometric function as an expression involving only a specified trigonometric function.  Although this is not necessarily easy, it is a common homework assignment in Trigonometry, so it should not be too terribly difficult to do here.  In fact, it would be good practice to work through this method for each of the other inverse trigonometric functions.  (Be sure to pay close attention to the range of each inverse trigonometric function and how it affects the result.)