### All Calculus AB Resources

## Example Questions

### Example Question #1 : Differentiate Inverse Functions

Which expression correctly identifies the inverse of ?

**Possible Answers:**

**Correct answer:**

The inverse of a function can be found by substituting yvariables for the variables found in the function, then setting the function equal to . By next isolating , the inverse function is written. Then, the notation is used to describe the newly written function as being the inverse of the original function. The answer choice “” is correct.

### Example Question #1 : Differentiate Inverse Functions

Which of the following correctly identifies the derivative of an inverse function?

**Possible Answers:**

**Correct answer:**

This question asks you to recognize the correct notation of a differentiating inverse functions problem. First, it is key to recognize that the equation needs to have the same variable throughout, thus eliminating the answer choices and . Next, there should be no constants in the correct equation; thus, is incorrect. The correct choice is .

### Example Question #1 : Differentiate Inverse Functions

Find given .

**Possible Answers:**

**Correct answer:**

Let

It is important to recognize the relationship between a function and its inverse to solve.

If , solving for the inverse function will produce .

To find the derivative of an inverse function, use:

Therefore,

### Example Question #2 : Differentiate Inverse Functions

Let .Find .

**Possible Answers:**

**Correct answer:**

To find the derivative of the inverse of , it is useful to first solve for .

This will help because is needed in the derivative equation, .

Next, the equation for the derivative of an inverse function can be evaluated.

After substituting in for , the derivative of or , is applied:

Therefore, the correct answer is

### Example Question #1 : Differentiate Inverse Functions

Let . Find .

**Possible Answers:**

**Correct answer:**

To find the derivative of the inverse of , it is useful to first solve for .

This will help because is needed in the derivative equation, .

Next, the equation for the derivative of an inverse function can be evaluated.

After substituting in , the derivative of , or, is applied:

Therefore, the correct answer is

### Example Question #2 : Differentiate Inverse Functions

Let . Find .

**Possible Answers:**

**Correct answer:**

To find the derivative of the inverse of , it is useful to first solve for .

This will help because is needed in the derivative equation, .

Next, the equation for the derivative of an inverse function can be evaluated.

After substituting in for , the derivative of , or , is applied:

Therefore, the correct answer is .

### Example Question #3 : Differentiate Inverse Functions

Let . Find .

**Possible Answers:**

**Correct answer:**

To find the derivative of the inverse of , it is useful to first solve for .

This will help because is needed in the derivative equation, .

Next, the equation for the derivative of an inverse function can be evaluated.

After substituting in for , the derivative of , or , is applied:

Therefore, the correct answer is .

### Example Question #1 : Differentiate Inverse Functions

Let . Find .

**Possible Answers:**

**Correct answer:**

To find the derivative of the inverse of , it is useful to first solve for .

This will help because is needed in the derivative equation, .

Next, the equation for the derivative of an inverse function can be evaluated.

After substituting in for , the derivative of , or , is applied:

Therefore, the correct answer is

### Example Question #2 : Differentiate Inverse Functions

Suppose the points in the table below represent the continuous function . The differentiable function is the inverse of the function . Find .

**Possible Answers:**

**Correct answer:**

Below is the equation for the derivative of :

So, the value of must first be found.

Using the data from the table, since .

Next, from the table the following can be obtained:

Now, the appropriate substitutions can be made to solve for .

### Example Question #1 : Differentiate Inverse Functions

Find given .

**Possible Answers:**

**Correct answer:**

To find the derivative of the inverse of , it is useful to first solve for .

This will help because is needed in the derivative equation, .

Next, the equation for the derivative of an inverse function can be evaluated.

After substituting in for , the derivative of , or , is found by taking the derivative of and applying chain rule.

After finding the general term , evaluate at .

Therefore, the correct answer is .

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