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Lesson 19 of 24
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# Derivatives of Inverse Functions

In this section we explore the relationship between the derivative of a function and the derivative of its inverse. For functions whose derivatives we already know, we can use this relationship to find derivatives of inverses without having to use the limit definition of the derivative. In particular, we will apply the formula for derivatives of inverse functions to trigonometric functions. This formula may also be used to extend the power rule to rational exponents.

# 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. (Figure) shows the relationship between a function  and its inverse . Look at the point  on the graph of  having a tangent line with a slope of . This point corresponds to a point  on the graph of  having a tangent line with a slope of . 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 . Then by differentiating both sides of this equation (using the chain rule on the right), we obtain.

Solving for , we obtain.

We summarize this result 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 ,.

Alternatively, if  is the inverse of , then.

### Applying the Inverse Function Theorem

Use the Inverse Function Theorem to find the derivative of . Compare the resulting derivative to that obtained by differentiating the function directly.

#### Solution

The inverse of  is . Since , begin by finding . Thus, and .

Finally,.

We can verify that this is the correct derivative by applying the quotient rule to  to obtain.

Use the inverse function theorem to find the derivative of  Compare the result obtained by differentiating  directly.[reveal-answer q=”336869″]Show Solution[/reveal-answer]

Hint

Use the preceding example as a guide.

### Applying the Inverse Function Theorem

Use the inverse function theorem to find the derivative of .

#### Solution

The function  is the inverse of the function . Since , begin by finding . Thus, and .

Finally,.

Find the derivative of  by applying the inverse function theorem.

#### Hint

Use the fact that  is the inverse of .

From the previous example, we see that we can use the inverse function theorem to extend the power rule to exponents of the form , where  is a positive integer. This extension will ultimately allow us to differentiate , where  is any rational number.

### Extending the Power Rule to Rational Exponents

The power rule may be extended to rational exponents. That is, if  is a positive integer, then.

Also, if  is a positive integer and  is an arbitrary integer, then.

## Proof

The function  is the inverse of the function . Since , begin by finding . Thus, and .

Finally,.

To differentiate  we must rewrite it as  and apply the chain rule. Thus,

### Applying the Power Rule to a Rational Power

Find the equation of the line tangent to the graph of  at .

#### Solution

First find  and evaluate it at . Since and

the slope of the tangent line to the graph at  is .

Substituting  into the original function, we obtain . Thus, the tangent line passes through the point . Substituting into the point-slope formula for a line and solving for , we obtain the tangent line.

Find the derivative of .

#### Hint

Rewrite as  and use the chain rule.

# 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 functions. Previously, derivatives of algebraic functions have proven to be algebraic functions and derivatives of trigonometric functions have been shown to be trigonometric functions. Here, for the first time, we see that the derivative of a function need not be of the same type as the original function.

### Derivative of the Inverse Sine Function

Use the inverse function theorem to find the derivative of .

#### Solution

Since for  in the interval  is the inverse of , begin by finding . Since and ,

we see that.

#### Analysis

To see that , consider the following argument. Set . In this case,  where . We begin by considering the case where . Since  is an acute angle, we may construct a right triangle having acute angle , a hypotenuse of length 1, and the side opposite angle  having length . From the Pythagorean theorem, the side adjacent to angle  has length . This triangle is shown in (Figure). Using the triangle, we see that .

In the case where , we make the observation that  and hence .

Now if  or  or , and since in either case  and , we have.

Consequently, in all cases, .

### Applying the Chain Rule to the Inverse Sine Function

Apply the chain rule to the formula derived in (Figure) to find the derivative of  and use this result to find the derivative of .

#### Solution

Applying the chain rule to , we have.

Now let , so . Substituting into the previous result, we obtain

Use the inverse function theorem to find the derivative of .

#### Hint

The inverse of  is . Use (Figure) as a guide.

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

### Applying Differentiation Formulas to an Inverse Tangent Function

Find the derivative of .

#### Solution

Let , so . Substituting into (Figure), we obtain.

Simplifying, we have.

### Applying Differentiation Formulas to an Inverse Sine Function

Find the derivative of .

#### Solution

By applying the product rule, we have.

Find the derivative of .

#### Hint

Use (Figure). with

### Applying the Inverse Tangent Function

The position of a particle at time  is given by  for . Find the velocity of the particle at time .

#### Solution

Begin by differentiating  in order to find . Thus,.

Simplifying, we have.

Thus, .

Find the equation of the line tangent to the graph of  at .

#### Hint

gives the slope of the tangent line.

### 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.
• Power rule with rational exponents
.
• 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