Calculus 3 : Applications of Partial Derivatives

Example Questions

Example Question #71 : Applications Of Partial Derivatives

Find  of the following function:

Explanation:

The gradient of the function is

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #72 : Applications Of Partial Derivatives

Find  for the following function:

Explanation:

The gradient of a function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The rules used to find the derivatives are

Example Question #41 : Gradient Vector, Tangent Planes, And Normal Lines

Find  of the following function:

Explanation:

The gradient of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #74 : Applications Of Partial Derivatives

Find  of the given function:

Explanation:

The gradient of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #75 : Applications Of Partial Derivatives

Find of the following function:

Explanation:

The gradient of the function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #71 : Applications Of Partial Derivatives

Find the equation of the tangent plane to the following function at :

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rule:

Evaluated at the given point, the partial derivatives are

Note that the partial derivative with respect to z was 4 to begin with; the fact that the point has a z coordinate of 4 is a coincidence.

Now, plug all of this into our given formula:

which simplified becomes

Example Question #77 : Applications Of Partial Derivatives

Find  of the following function:

Explanation:

The gradient of a function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #78 : Applications Of Partial Derivatives

Find  of the following function:

Explanation:

The gradient of a function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #79 : Applications Of Partial Derivatives

Find  of the following function:

Explanation:

The gradient of a function is given by

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The derivatives were found using the following rules:

Example Question #80 : Applications Of Partial Derivatives

Find the equation of the tangent plane to the given function at :

Explanation:

The equation of the tangent plane is given by

So, we must find the partial derivatives for the function evaluated at the point given. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

The partial derivatives evaluated at the given point are

Plugging all of this into the above formula, we get