# Calculus 3 : Partial Derivatives

## Example Questions

### Example Question #12 : Differentials

Find the differential of the function:

Explanation:

The differential of a function  is given by

The partial derivatives of the function are

### Example Question #13 : Differentials

Find the differential of the function:

Explanation:

The differential of the function  is given by

The partial derivatives are

### Example Question #14 : Differentials

Find the total differential, , of the following function

Explanation:

The total differential is defined as

For the function

We first find

by taking the derivative with respect to  and treating  as a constant.

We then find

by taking the derivative with respect to  and treating  as a constant.

We then substitute these partial derivatives into the first equation to get the total differential

### Example Question #15 : Differentials

Find the total differential, , of the following function

Explanation:

The total differential is defined as

For the function

We first find

by taking the derivative with respect to  and treating  as a constant.

We then find

by taking the derivative with respect to  and treating  as a constant.

We then substitute these partial derivatives into the first equation to get the total differential

### Example Question #16 : Differentials

Find the total differential, , of the following function

Explanation:

The total differential is defined as

For the function

We first find

by taking the derivative with respect to  and treating  as a constant.

We then find

by taking the derivative with respect to  and treating  as a constant.

We then substitute these partial derivatives into the first equation to get the total differential

### Example Question #17 : Differentials

If , calculate the differential  when moving from  to.

Explanation:

The equation for  is

.

Evaluating partial derivatives and substituting, we get

Plugging in, we get

.

### Example Question #18 : Differentials

If , calculate the differential  when moving from the point  to the point .

Explanation:

The equation for  is

.

Evaluating partial derivatives and substituting, we get

Plugging in, we get

.

### Example Question #19 : Differentials

If , calculate the differential  when moving from the point  to the point .

Explanation:

The equation for  is

.

Evaluating partial derivatives and substituting, we get

Plugging in, we get

### Example Question #20 : Differentials

Find the differential of the following function:

Explanation:

The differential of the function is given by

The partial derivatives are

### Example Question #1440 : Partial Derivatives

Find the differential of the following function: