# Trig Inversesap Calculus

## Inverse Function Review

1. Practice the relationship between the derivatives of inverse functions: if f and g are inverse functions, then f’(x)=1/(g’(f(x)).
2. Math AP®︎/College Calculus AB Differentiation: composite, implicit, and inverse functions Differentiating inverse trigonometric functions Differentiating inverse trig functions review AP.CALC.
3. Chapter 4 - Trigonometric and Inverse Trigonometric Functions. Maxima and Minima Using Trigonometric Functions; Problems in Caculus Involving Inverse Trigonometric Functions. 37 - A ladder sliding downward; 38 - Rate of rotation of search light pointing to a ship; 39 - Rate of increase of angle of elevation of the line of sight.
4. Teaching Yourself is the best way to learn. APCalc Lesson 14 — Derivative Rules and Laws.

Derivatives of Inverse Trigonometric Functions. We now turn our attention to finding derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text. The derivatives of inverse trigonometric functions are quite surprising in that their derivatives are actually algebraic.

One application of the chain rule is to compute the derivative of an inverse function. First, let's review the definition of an inverse function:

We say that the function is invertible on an interval [a, b] if there are no pairsin the interval such thatand . That means there are no two x-values that have the same y-value. That's important, because if two x-coordinates map to the same y-coordinate, the inverse function (working in reverse) would map a single x-coordinate to multiple y-coordinates. That doesn't make sense, because f(x) could have more than one resulting value!

We say thatis the inverse of an invertible functionon [a, b] if:

For example, the functionsandare inverses onsinceon that interval. Note that it works both ways -- the inverse function of the original function returns x, and the original function performed on the inverse ALSO returns x.

### Taking The Derivative

So, how do we differentiate an inverse function? Recall the chain rule:

### Trig Inversesap Calculus Problems

Applying this to the definition of an inverse function, we have:

So:

Let's see how to apply this to real examples.

#### Example 1

Letsoas above. Then, and applying the formula we have:

This agrees with the answer we would get from viewingas the polynomial function.

#### Example 2

The functionis invertible on the interval, with inverse. We know that, so applying our formula we see that

### Trig Inversesap Calculus Formula

We can check that , which means that

.