# AP Calculus AB : Implicit differentiation

## Example Questions

### Example Question #11 : Implicit Differentiation

Given that , find the derivative of the following function:

Explanation:

To find the derivative of the function, we use implicit differentiation, where we always treat y as a function of x, and denoting any derivative of y with respect to x as

### Example Question #12 : Implicit Differentiation

Find  from the following equation:

Explanation:

To find the derivative of y with respect to x, we must take the derivative with respect to x of both sides of the equation:

The derivatives were found using the following rules:

Notice the chain rule was used for every function containing y, because y is a function of x and its whose derivative we are interested in isolating.

Using algebra to solve, our final answer is

### Example Question #13 : Implicit Differentiation

Find :

Explanation:

To find  we must use implicit differentiation, which is an application of the chain rule.

Taking  of both sides of the equation, we get

The derivatives were found using the following rules:

Note that for every derivative of a function with y, the additional term  appears; this is because of the chain rule, where y=g(x), so to speak, for the function it appears in.

Solving for , we get

### Example Question #14 : Implicit Differentiation

Evaluate  at the point (1, 4) for the following equation:

.

Explanation:

Using implicit differentiation, taking the derivative of the given equation yields

Getting all the  's to their own side, we have

Factoring,

And dividing

Plugging in our point, , we have

### Example Question #15 : Implicit Differentiation

Determine :

Explanation:

To find  we must use implicit differentiation, which is an application of the chain rule.

Taking the derivative with respect to x of both sides of the equation, we get

The derivatives were found using the following rules:

Note that for every derivative of a function with y, the additional term  appears; this is because of the chain rule, where , so to speak, for the function it appears in.

Using algebra to solve for , we get

### Example Question #16 : Implicit Differentiation

Find :

Explanation:

To find  we must use implicit differentiation, which is an application of the chain rule.
Taking  of both sides of the equation, we get

which was found using the following rules:

Note that for every derivative of a function with y, the additional term appears; this is because of the chain rule, where , so to speak, for the function it appears in.

Using algebra to solve for , we get

### Example Question #17 : Implicit Differentiation

Given that , compute the derivative of the following function

Explanation:

To find the derivative of the function, we use implicit differentiation, which is an application of the chain rule. We use this because , and any derivative with respect to  is  (or ).

First, we use the chain rule combined with the product rule in taking the derivative of y

Then we expand in order to isolate the terms with

Then we factor out a

### Example Question #18 : Implicit Differentiation

Given that , compute the derivative of the following function:

Explanation:

To find the derivative of the function, we use implicit differentiation, which is an application of the chain rule. We use this because , and any derivative with respect to  is  (or ).

First, we use the chain rule combined with the product rule in taking the derivative of y

Then isolate the terms with

Then we factor out a

### Example Question #19 : Implicit Differentiation

Given that , compute the derivative of the following function:

Explanation:

To find the derivative of the function, we use implicit differentiation, which is an application of the chain rule. We use this because , and any derivative with respect to  is  (or ).

First, we use the chain rule combined with the product rule in taking the derivative of y

Then, we expand in order to isolate the terms with

Finally, we factor out a

### Example Question #20 : Implicit Differentiation

Given that , find the derivative of the function using implicit differentiation