3.2 Partial Derivatives

It isn't hard to imagine functions that depend on more than a single independent variable.

$$p = \frac{nRT}{V}$$ (3.2.4)

is such a function with p depending on the input values of n, T, and V.

Symbolically this is (using only two independent variables)

$$z = f(x,y)$$ (3.2.5)

a typical example being:

$$z = x^2 y - e^y$$ (3.2.6)

Again, z will certainly change if x and y are changed. How do we calculate that change?

Let's formulate that question more precisely. By how much will z in equation 3.2.3 change if we change x by a very small amount? We aren't going to change y at all.

The answer is that we figure it out just as we did in "ordinary" calculus. We differentiate equation 3.2.3 with respect to x and get the result

$$2xy$$

just as we ordinarily would. Note that we pretend that y is a constant during this process-because it does not change in this process.

This is called a partial derivative. In particular it is the partial derivative of z with respect to x holding y constant. There's a notation for this that uses a strange symbol:

$$\left( \frac{\partial z}{\partial x} \right)_y$$

Of course we could partially differentiate z with respect to y too. The result is:

$$\left( \frac{\partial z}{\partial y} \right)_x = x^2 - e^y$$ (3.2.7)

just as it would be if x were a "real" constant.

Partial derivatives come up very often in physical chemistry since most things depend on more than one variable. Since they are so closely related to ordinary derivatives, they exist under the same conditions of continuity, etc.