# AP Calculus AB : Implicit differentiation

## Example Questions

### Example Question #74 : Applications Of Derivatives

Differentiate the following implicit function:

Explanation:

For this problem we are asked to find , or the rate of change in y with respect to x.

To do this we take the derivative of each variable and to differentiate between the two, we will write dx or dy after.

would then become

We note that the derivative of a constant is still zero.

We must now rewrite this function in the form

### Example Question #75 : Applications Of Derivatives

Find the implicit derivative,  a circle centered at  with radius .

Explanation:

The equation of a circle centered at  with radius  is .

We first expand our equation to simplify the derivative.

Take the derivatives of x and y we get:

Since the derivative of a constant is zero.

Next we must rewrite our equation in terms of :

Simplifying:

### Example Question #76 : Applications Of Derivatives

Given that , find the derivative of the function

Explanation:

To find the derivative with respect to y, we use implicit differentiation, which is an application of the chain rule.

### Example Question #77 : Applications Of Derivatives

Given that , find the derivative of the function

Explanation:

To find the derivative with respect to y, we use implicit differentiation, which is an application of the chain rule

### Example Question #78 : Applications Of Derivatives

Given that , find the derivative of the function

Explanation:

To find the derivative with respect to y, we use implicit differentiation, which is an application of the chain rule

### Example Question #79 : Applications Of Derivatives

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 derivative 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 #80 : Applications Of Derivatives

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 following derivative rules were used:

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.

Using algebra to solve for , we get

### Example Question #81 : Applications Of Derivatives

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

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.

Using algebra to solve for , we get

.

### Example Question #82 : Applications Of Derivatives

Use implicit differentiation to calculate the equation of the line tangent to the equation  at the point (2,1).

Explanation:

Differentiate both sides of the equation:

Simplify:

Use implicit differentiation to differentiate the y term:

Subtract 4x from both sides of the equation:

Divide both sides of the equation by 2y:

Plug in the appropriate values for x and y to find the slope of the tangent line:

Use slope-intercept form to solve for the equation of the tangent line:

Plug in the appropriate values of x and y into the equation, to find the equation of the tangent line:

Solve for b:

Solution:

### Example Question #83 : Applications Of Derivatives

Find , where  is a function of x.

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

and the derivatives were found using the following rules:

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

Using algebra to solve, we get