# Calculus 3 : Partial Derivatives

## Example Questions

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### Example Question #1441 : Partial Derivatives

Find the differential of the following function:

Explanation:

The differential 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 #1442 : Partial Derivatives

Find the differential of the following function:

Explanation:

The differential 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 #1443 : Partial Derivatives

Find the total differential of the following function:

Explanation:

The total differential 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 #1444 : Partial Derivatives

Find the total differential of the function:

Explanation:

The total differential 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 #1445 : Partial Derivatives

If , calculate the total differential .

Explanation:

The total differential  of a function  is defined as the sum of the partial derivatives of  with respect to each of its variables; that is,

In this case, , and so we use the sum rule, the rule for derivatives of a variable raised to a power, the rule for the derivative of , and the chain rule to calculate the partial derivatives  and , as shown:

,

.

In both cases, we treated the variable not being differentiated as a constant, and applied the chain rule to  to calculate its partial derivatives. Now that  and  have been calculated, all that remains is to substitute them into the definition of the total derivative:

### Example Question #1446 : Partial Derivatives

If , calculate the total differential .

Explanation:

The total differential  of a function  is defined as the sum of the partial derivatives of  with respect to each of its variables; that is,

In this case, , and so we use the product rule and the rule for differentiating  to calculate the partial derivatives  and , as shown:

,

In both cases, we treated the variable not being differentiated as a constant, and applied the product rule to calculate its partial derivatives. Now that  and  have been calculated, all that remains is to substitute them into the definition of the total derivative:

### Example Question #1447 : Partial Derivatives

Compute the differentials for the following function.

Explanation:

What we need to do is take derivatives, and remember the general equation.

When taking the derivative with respect to y recall that the product rule needs to be used.

### Example Question #1448 : Partial Derivatives

Compute the differentials for the following function.