Cracking The Ap Calculus ab Exam 2018

INVERSE TRIG FUNCTIONS

In this unit, you’ll learn a broader set of integration techniques. So far, you know how to do only a few
types of integrals: polynomials, some trig functions, and logs. Now we’ll turn our attention to a set of
integrals that result in inverse trigonometric functions. But first, we need to go over the derivatives of
the inverse trigonometric functions. (You might want to refer back to Chapter 12 on derivatives of inverse
functions.)

Suppose you have the equation sin y = x. If you differentiate both sides with respect to x, you get

cos y   =   1

Now divide both sides by cos y.

=

Because sin^2 y + cos^2 y = 1, we can replace cos y with .

=

Finally, because x = sin y, replace sin y with x. The derivative equals

Now go back to the original equation sin y = x and solve for y in terms of x: y = sin−1 x.

If you differentiate both sides with respect to x, you get

    =       sin−1 x

Replace , and you get the final result.

    sin−1 x =   

This is the derivative of inverse sine. By similar means, you can find the derivatives of all six inverse trig
functions. They’re not difficult to derive, and they’re also not difficult to memorize. The choice is yours.
We use u instead of x to account for the Chain Rule.