# Calculus 3 : Applications of Partial Derivatives

## Example Questions

### Example Question #11 : Gradient Vector, Tangent Planes, And Normal Lines

Calculate  given

Explanation:

By definition,

, where  are the respective  components of .

Therefore, we need to calculate the above terms, shown as

Therefore,

.

### Example Question #42 : Applications Of Partial Derivatives

Find , where

Explanation:

The gradient vector of f, , is equal to

So, we must find the partial derivatives of the function with respect to x, y, and z, keeping the other variables constant for each partial derivative:

The derivatives were found using the following rules:

Plugging this in to a vector, we get

### Example Question #43 : Applications Of Partial Derivatives

Find , where f is 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.

Now, we find the partial derivatives:

The derivatives were found using the following rules:

, , ,

### Example Question #44 : Applications Of Partial Derivatives

Find  of the following function:

Explanation:

To find the given partial derivative of the function, we must treat the other variable(s) as constants. For higher order partial derivatives, we work from left to right for the given variables.

To start, we must find the partial derivative with respect to y:

The following rules were used to find the derivative:

Next, we find the derivative of the above function above with respect to x:

The rule used is stated above.

Finally, find the partial derivative of the above function with respect to z:

The rule used is already stated above.

### Example Question #45 : Applications Of Partial Derivatives

Find  of the given function:

Explanation:

The gradient of a function is given by

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

The derivatives were found using the following rules:

### Example Question #41 : Applications Of Partial Derivatives

Find  of the 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 #47 : Applications Of Partial Derivatives

Find  for the given 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 #48 : 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 #49 : 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 derivatives were found using the following rules:

### Example Question #50 : Applications Of Partial Derivatives

Find  for the following function: