# Calculus 3 : Applications of Partial Derivatives

## Example Questions

### Example Question #61 : Applications Of Partial Derivatives

Find  of the function

Explanation:

To find the gradient vector, you use the following definition: . Taking the partial derivatives with respect to x, y, and z, we get , and . Putting these into the vector produces the right answer.

### Example Question #62 : Applications Of Partial Derivatives

Find  of the function

Explanation:

To find the gradient vector for a function , we use the definition

.

### Example Question #63 : Applications Of Partial Derivatives

Find  of the function

Explanation:

To find the gradient vector for a function , we use the definition

.

### Example Question #64 : Applications Of Partial Derivatives

Find  of the function

Explanation:

To find the gradient vector for a function , we use the definition

.

### Example Question #65 : Applications Of Partial Derivatives

Compute the gradient of the following scalar function:

Explanation:

The gradient of a function is defined as:

For our function:

### Example Question #66 : Applications Of Partial Derivatives

Compute the gradient of the following scalar function:

Explanation:

The gradient of a function is defined as:

For our function:

### Example Question #67 : 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

### Example Question #68 : 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 rules used to find the derivatives are

### Example Question #69 : Applications Of Partial Derivatives

Find  of the function

Explanation:

To find the gradient vector, you use the formula , where . Using the function given and the rules for partial differentiation, we e. Plugging these values into vector notation gets you the correct answer.

### Example Question #70 : Applications Of Partial Derivatives

Find  of the function