3.3 Total Derivatives

The very fact that is such a thing as a partial derivative should be a hint that there just might be a total derivative as well.

The total derivative answers the question "What happens to z is I change both x and y by small amounts at the same time?"

The "small amounts" thing is important since it makes things very simple. A function of many variables can be represented by a graph in one more dimension than there are independent variables in the function. The result is a surface whose "height" is the value of the dependent variable.

That surface will generally have mountains and valleys and be fairly complicated. However, if we look at only very very small areas of the surface, it will, to a first approximation, appear flat. Not flat meaning parallel to the "floor", but flat as in a piece of paper held at an angle.

This idea is important because on a flat surface I can vary the independent variables independently. I can treat a step in the x direction as being taken with y constant because it can't have changed by much if the step is very little. And the same is true of a step in the y direction.

In this approximation (which can be made exact) it does not matter if I first change x by a little bit and then change y by another small bit or do it in the reverse order. The result will be the same.

Mathematically, for a function of two independent variables this is:

$$dz = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy$$ (3.3.8)

Yes, it is that simple. But pay attention to the partial derivative signs and the ordinary derivative signs. The dx and dy mean an infinitesimal change in x and y, and dz is the resultant change in z.

The rules for calculating the total derivative involve just following the formula above. Here's an example:

$$z = x^2 \ln (y) + x e^y$$ (3.3.9)

Then

$$dz = ( 2x \ln (y) + e^y ) dx + \left( \frac{x^2}{y} + x e^y \right) dy$$ (3.3.10)

This extends to more than two variables by simply adding more terms.