# Partial Derivative: Definition, Rules & Examples

## Partial Derivative Definition

Calories consumed and calories burned have an impact on our weight. Let's say that our weight, *u*, depended on the calories from food eaten, *x*, and the amount of physical exertion we do, *y*. If we only regulated our eating, while doing the same exercise every day, we could ask how does *u* change when we vary only *x*. Likewise, we could keep *x* constant and take note of how *u* varies when we change *y*. This would be like keeping a constant daily diet while changing how much we exercise. This idea of change with respect to one variable while keeping other variables constant is at the heart of the **partial derivative**.

When we write *u* = *u*(*x*,*y*), we are saying that we have a function, *u*, which depends on two independent variables: *x* and *y*. We can consider the change in *u* with respect to either of these two independent variables by using the partial derivative. The partial derivative of *u* with respect to *x* is written as:

What this means is to take the usual derivative, but only *x* will be the variable. All other variables will be treated as constants. We can also determine how *u* changes with *y* when *x* is held constant. This is the partial of *u* with respect to *y*. It's written as:

For example, if *u* = *y* * *x*^2, then:

Likewise, we can differentiate with respect to *y* and treat *x* as a constant with the equation:

The **rule** for partial derivatives is that we differentiate with respect to one variable while keeping all the other variables constant. As another example, find the partial derivatives of *u* with respect to *x* and with respect to *y* for:

To do this example, we will need the derivative of an exponential with the following:

And the derivative of a cosine, which is written as:

Thus:

and

## Second Order Derivatives

So far we have defined and given examples for first-order partial derivatives. **Second-order partial derivatives** are simply the partial derivative of a first-order partial derivative. We can have four second-order partial derivatives, which you can see right here:

Continuing with our first example of *u* = *y* * *x*^2,

and

Likewise,

and

And in our second example:

and

The **mixed partial derivatives** become:

and

Note that we will always get:

This can be used to check our work.

## Extending Partial Derivatives

What if the variables *x* and *y* also depend on other variables? For example, we could have *x* = *x*(*s*,*t*) and *y* = *y*(*s*,*t*).

Then, as you can see, we get the partial derivatives through addition of the different partial derivatives of *x* and *y*.

Let's work this out given the following functions:

Find:

We first calculate the required first-order partial derivatives:

Then:

## Other Derivative Notations

Partial derivatives can be expressed using a subscript. In the case of first-order partial derivatives:

For second-order partial derivatives:

If you look carefully at each step in the following example, you will see why the order of the subscripts for mixed partial derivatives is reversed, which is reflected here:

## Lesson Summary

Let's very briefly review what we've learned about partial derivatives. First, we saw that **partial derivatives** are evaluated by treating one variable as the independent variable while keeping all other variables constant. We can take first-order partial derivatives by following the rules of ordinary differentiation.

Then we looked at how **second-order partial derivatives** are partial derivatives of first-order partial derivatives. We next saw how to evaluate partial derivatives when the variables are dependent on other variables. Finally, we looked at a subscript notation for expressing the partial derivative, both with the first and second orders.

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## Partial Derivative Practice Questions

**1.** The function *f(x, y)* gives us the profit (in dollars) of a certain commodity as the number of commodities *x* sold and the number of days *y* the commodity is on the market. The function is below:

*f(x, y) = (x^2) e^(-y)*.

Find the rate of change of the profit with respect to the number of commodities sold and the number of days the commodity stays on the market.'

**2.** The following function below is a product of a logarithmic function and a trigonometric function. Find its first-order, partial derivatives:

*g(x, y) = ln(x) cos(4y)*.

### Answers

**1.** To find the rate of change of the profit, *f*, with respect to the number commodities sold, *x*, we take the partial derivative of *f* with respect to *x* while keeping *y* as constant. This yields the following:

*f_x = e^(-y) (x^2)' = e^(-y) 2x = 2x e^(-y)* dollars per number of commodities sold.

Similarly, to find the rate of change of the profit, *f*, with respect to the number of days, *y*, that the commodity stays on the market, we calculate the partial derivative of *f* with respect to *y* while keeping *x* as constant. This yields the following:

*f_y = [e^(-y)]' (x^2) = -e^(-y) (x^2) = -(x^2) e^(-y)* dollars per day.

**2.** To get the first-order, partial derivative of *g(x, y)* with respect to *x*, we differentiate *g* with respect to *x*, while keeping *y* constant. This leads to the following, first-order, partial derivative:

*g_x = [ln(x)]' cos(4y) = (1/x) cos(4y)*.

Similarly, to get the first-order, partial derivative of *g(x, y)* with respect to *y*, we differentiate *g* with respect to *y*, while keeping *x* constant. This leads to the following, first-order, partial derivative:

*g_y = ln(x) [cos(4y)]' = ln(x) -4 sin(4y) = -4 ln(x) sin(4y)*.

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