Partial Differentiation - University of Florida



Partial Differentiation

So first of all what is a partial derivative and why do we need it? A partial derivative is needed in functions with more than one variable. You can think of this in the context of a chemistry experiment. In order to run the test there are several parameters (or variables) that need to be monitored including temperature and pressure. So what if we want to look at the rate of change of the outcome of the experiment by only changing one variable? This is in essence the art of partial derivatives. We hold one independent variable constant and take the derivative with respect to the other variable. The notation takes a little getting used to, but the ideas are similar to the original differentiation method you learned in your first calculus class.

Say we have a function [pic] where the two independent variables are x and y. We find the partial derivatives as follows

The partial derivative with respect to x: [pic]

So we treat y as a constant and merely differentiate with respect to x. Notice the use of [pic]instead of d. [pic] is always used for partial differentiation.

The partial derivative with respect to y: [pic]

This time we hold x constant and differentiate with respect to y.

[pic]and [pic] are known as the first partials of z.

Example 1:

Find the first partials of the function [pic] and evaluate them at the point (1, ln2).

We start by finding the partial with respect to x.

[pic]

where the chain rule, exponential rule, and the multiplication rule have been used.

So, [pic]

Now, the partial with respect to y is given by

[pic]

Remember, for the y partial, we treat x as a constant (if this confuses you, just put a number there and keep track of it [don’t multiply to other constants] to remind yourself not to treat it as a variable).

Example 2: Finding slopes of a Surface in both the x and y directions

Find the slopes of the surface given by [pic] at the point (1,2,1) in the x-direction and the y-direction.

In English, this means we should find both partial derivatives and evaluate them at the given point (x, y, z). Think of the partial derivative like finding the rate of change with respect to one variable, so in other words this like finding the slope in a particular direction (curves are 3-D and have multiple slopes depending on which direction you look at it).

So to find the slope in the x direction, we find the first partial with respect to x.

[pic]

This indicates that at this point the curve is horizontal with respect to the x direction.

Now, the other case

[pic]

Again the slope in the y direction is also 0. If you could imagine what this curve looked like, it would resemble an upside down parabola shaped surface (bowl shaped) and we are at the top of this parabola (the maximum point).

Now, what about higher order partial derivatives?

This is where things get more complicated notation wise.

We define the following second order partial derivatives:

[pic]

The first two definitions are second partials with respect to x and y respectively. The third and fourth definitions are denoted as mixed partial derivatives. To remember what order you differentiate, look to which variable is closer to the “f.” The closer one to this is the first variable you differentiate with respect to. Although not proved here,

[pic]

So you really only need to find one mixed partial to know what the other one is. The reader is encouraged to prove this point for himself.

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