Suppose u is a function of x and y, as in the previous section. Now consider what happens to u if x and y increase by a small amount, say dx and dy respectively. The quantities dx and dy are called small increments.
We might expect that if x and y increase by a small amount, then u will also change by a small amount. Let's call that du.
The question is, can we say what du is, in terms of dx and dy?
Well, to see what effect such small changes will have on u, i.e. to find out what du is, we need to think back to what differentiation means.
The derivative of u with respect to x really means "the change in u divided by the change in x, as the change in x becomes infinitessimally small". In maths:
where dx is the change in x and du is the corresponding change in u.
Now that means that as dx gets smaller and smaller, the ratio du/dx gets closer and closer to being equal to the derivative.
Here's the crucial step. When dx has any small value, we can approximate the ratio du/dx by the derivative, i.e. we can say that ratio is approximately equal to the derivative.
So we have the following approximate relationship:
which holds only when dx is small. We can now rearrange this relationship to get an approximation for du!
That has all been very theoretical, so let's look at a particular example to make it clearer.
Suppose u is the volume of a cube, whose sides are of length x. Then u(x) is given by u(x)=x3.
Let's suppose the length x is 3cm and so the volume is 27cm3.
Now, what happens to the volume if x increases by 5mm?
Using the equation above, we can find the corresponding increase in volume, du:
so
so
and that's it!
We can do just the same with functions of more than one variable, so if u is a function of x and y, we find that:
