Inverse Trigonometric Functions (Alternate Method)

There is another method for finding the derivative of an inverse trigonometric function.  This alternate method uses a geometric approach that can be useful when setting up problems for analysis.  The use of this approach will require you to think in terms of right triangles, so we will be drawing pictures as we go.

Let’s again consider the arcsine function.  Let .  From this, we know several things.  First, we know .  Second, we know that .  Third, we know that .  Last, we know that if we draw a right triangle in which  is an angle that .  If  or , we will not have a triangle possible, so our picture will assume that .  Because , we have no difficulty representing both the positive and negative cases with a single triangle.

Note:  We used the Pythagorean Theorem to compute the third side.

Now, we can consider finding the derivative of the arcsine function from our previous result and use the groundwork we have laid here.  The only other thing we have to note is that .

We do in fact arrive at the same result but without the necessity of manipulating trigonometric expressions using trigonometric identities.  You might find it helpful to go back and find expressions for the derivatives of the other inverse trigonometric functions using this technique.  You can also verify that the results using this technique agree with your previous results.