# GRE Subject Test: Math : Partial Differentiation

## Example Questions

### Example Question #1 : Implicit Differentiation

Differentiate the following with respect to

Explanation:

The first step is to differentiate both sides with respect to :

Note: Those that are functions of  can be differentiated with respect to , just remember to mulitply it by

Now we can solve for :

### Example Question #1 : Partial Differentiation

Find

for .

Explanation:

Our first step would be to differentiate both sides with respect to :

The functions of  can be differentiated with respect to , just remember to multiply by  .

### Example Question #21 : Derivatives & Integrals

Differentiate the following to solve for .

Explanation:

Our first step is to differentiate both sides with respect to :

The functions of  can by differentiated with respect to , just remember to multiply them by

### Example Question #1 : Partial Differentiation

Differentiate the following with respect to .

Explanation:

Our first step is to differentiate both sides with respect to :

Note: we can differentiate the terms that are functions of  with respect to , just remember to multiply it by .

Note: The product rule was applied above:

### Example Question #1 : Partial Differentiation

Solve for :

Explanation:

To solve for the partial derivative, let all other variables be constants besides the variable that is derived with respect to.

In , the terms  are constants.

Derive as accordingly by the differentiation rules.

### Example Question #1 : Functions Of More Than Two Variables

Suppose the function .  Solve for .

Explanation:

Identify all the constants in function .

Since we are solving for the partial differentiation of variable , all the other variables are constants.  Solve each term by differentiation rules.

### Example Question #1 : Functions Of More Than Two Variables

Suppose the function .  Solve for .