### Learning Objectives

- Calculate the derivative of an inverse function.
- Recognize the derivatives of the standard inverse trigonometric functions.

In this section, we explore the relationship between the derivative of a function and the derivative of its inverse. We will use this relationship to find derivatives of inverse trigonometric functions.

## The Derivative of an Inverse Function

We begin by considering a function and its inverse. If is both invertible and differentiable, it seems reasonable that the inverse of is also differentiable. Let us look at the graphs of a function and its inverse on Figure 1 below. Consider the point on the graph of having a tangent line with a slope of . As we discussed in the previous section, the graphs of and are symmetric with respect to the line . Therefore, the tangent line to the curve at the point must be symmetric to the tangent line to the curve at the (symmetric) point . Note that the product of slopes of the lines that are symmetric with respect to the line is 1. Indeed, if a line has equation , symmetric one would have equation (switching and ), which is equivalent to , provided . Thus, if is differentiable at , then it must be the case that

We may also derive the formula for the derivative of the inverse by first recalling that for every in the domain of . Then by differentiating both sides of this equation (using the chain rule on the right), we obtain

.

Solving for , we obtain

.

We summarize the above in the following theorem.

### Inverse Function Theorem

Let be a function that is both invertible and differentiable. Let be the inverse of . For all satisfying ,

.

In other words, if we let be the inverse of , then

whenever .

## Derivatives of Inverse Trigonometric Functions

We now use the inverse function theorem to find derivatives of inverse trigonometric functions. These derivatives will prove invaluable in the study of integration later in this text.

### Derivative of the Inverse Sine Function

Use the inverse function theorem to find the derivative of .

#### Solution

Since is differentiable and invertible when restricted to the interval , as per the above theorem, for , we have that whenever . Because , we need to compute .

Let . Then , and we want to find . Since on , it follows from the Pythagorean trigonometric identity that .

The domain of is and for every . Therefore,

Use the inverse function theorem to find the derivative of .

#### Answer

The derivatives of the remaining inverse trigonometric functions may also be found by using the inverse function theorem. The corresponding formulas are provided in the following theorem.

### Derivatives of Inverse Trigonometric Functions

### Applying Formulas for the Derivatives of Inverse Trigonometric Functions

Find the derivatives of the following functions.

#### Solution

- We apply the chain rule with outside function and inside function to obtain
- Here we need to apply the product rule to the multiples and , while differentiating requires using the chain rule. We have

### Key Concepts

- The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative.
- We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.

## Key Equations

**Inverse function theorem**

whenever and is differentiable.**Derivative of inverse sine function**

**Derivative of inverse cosine function**

**Derivative of inverse tangent function**

**Derivative of inverse cotangent function**

**Derivative of inverse secant function**

**Derivative of inverse cosecant function**

.

### Exercises

For the following exercises, find .

** 1. **

#### Answer

** 2. **

** 3. **

#### Answer

**4. **

** 5. **

#### Answer

** 6. **

** 7. **

#### Answer

** 8. **

** 9. **

#### Answer

** 10. **

For the following exercises, find the slope of the tangent line to the given curve at the given point.

**11. ** at .

#### Answer

**12.**at the point corresponding to .

**13*.** at .

(*Hint: *Use implicit differentiation.)

#### Answer

**14*. ** There is a theorem that if a function is differentiable on an open interval and on then is constant on . Using this result, prove that , .

**15. [T]** A pole stands 75 feet tall. An angle is formed when wires of various lengths (measured in feet) are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle with respect to the wire length when a wire of length 90 feet is attached.

#### Answer

radians per foot

** 16. [T]** A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. After launch, let be the height of the rocket and be the angle of elevation of the camera. Find the rate of change of the angle of elevation with respect to the rocket’s height when the camera and the rocket are 5000 feet apart.

** 17*. **A local movie theatre has a 30-foot-high screen that is 10 feet above a person’s eye level when seated. Suppose that a person is sitting at a distance of feet from the movie screen and has a viewing angle of radians, see the figure below.

- Find .
- Use optimization methods learned in Calculus 1 to determine at what distance the person should sit to maximize his or her viewing angle.

#### Answer

a.

b. The optimal distance for maximizing the viewing angle is 20 feet.