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Partial Differentiation: Definition, Rules & Application

Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

In this lesson, you will be introduced to a method for finding derivatives of a multivariable function, the so-called partial derivatives. A few examples and applications will also be given.

The Definition of the Partial Derivative

If you know how to take a derivative, then you can take partial derivatives. Suppose f is a multivariable function, that is, a function having more than one independent variable, x, y, z, etc. The partial derivative with respect to a given variable, say x, is defined as taking the derivative of f as if it were a function of x while regarding the other variables, y, z, etc., as constants. For example, if f is a function of x, y, and z, then there are three different partial derivatives for f -- one with respect to x, one with respect to y, and one with respect to z.

Common notations for the partial derivatives include the following (here, we are looking at a function of two variables, but the notations are similar for any number of variables).


Notations for partial derivatives
Partial derivative notations


A Basic Example

Let's find the partial derivatives of z = f(x, y) = x^2 sin(y). This function has two independent variables, x and y, so we will compute two partial derivatives, one with respect to each variable.

  1. The partial derivative of f with respect to x is 2x sin(y). Since we are treating y as a constant, sin(y) also counts as a constant. Thus, the only thing to do is take the derivative of the x^2 factor (which is where that 2x came from).
  2. The partial derivative of f with respect to y is x^2 cos(y). This time, we treat x (and hence also x^2) as a constant, and simply take the derivative of sin(y).

That's really all there is to it! Now let's explore what the partial derivatives are good for.

Interpretation as a Rate of Change

Recall from calculus, the derivative f '(x) of a single-variable function y = f(x) measures the rate at which the y-values change as x is increased. The more steeply f increases at a given point x = a, the larger the value of f '(a).

So what happens when there is more than one variable? Let's look at the two-varible case, z = f(x, y). The partial derivative of f with respect to x measures the rate at which z-values change as x is increased while y is held constant. Similarly, the partial derivative of f with respect to y measures the rate at which z-values change as y is increased while x is held constant. Confused? Perhaps a concrete example may clarify.

Let's suppose you're an avid hiker and you are currently trekking over some rough terrain with lots of hills and valleys. Let's call east the positive x direction, and north the positive y direction. Now when you set off from your location at some point (a, b), you might have to climb a hill as you go east. This would correspond to a positive value for the partial derivative with respect to x evaluated at the point (a, b). On the other hand, if you turned north instead, it may be that you can descend into a valley. This would give a negative value for the partial derivative with respect to y evaluated at (a, b). Partial derivatives are the mathematical tools used to measure increase or decrease with respect to a particular direction of travel.

Very bumpy terrain. The values of the partial derivatives at any given point indicate whether you are traveling up a hill or down a valley.
Bumpy terrain

More Examples

Let's get some practice finding the partial derivatives of a few functions. Remember, all of the usual rules and formulas for finding derivatives still apply - the only new thing here is that one or more variables must be considered constant.

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