Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/220

 In the same way, if

$$u = f(x,\ y,\ z),$$

and $$x$$, $$y$$, $$z$$ are all functions of $$t$$, we get

and so on for any number of variables.

In (51) we may suppose $$t = x$$; then $$y$$ is a function of $$x$$, and $$u$$ is really a function of the one variable $$x$$, giving

In the same way, from (52) we have

The student should observe that $$\tfrac{\partial u}{\partial x}$$ and $$\tfrac{du}{dx}$$ have quite different meamngs. The partial derivative $$\tfrac{\partial u}{\partial x}$$ is formed on the supposition that the particular variable x alone varies, while

$$\frac{du}{dx} = \lim_{\Delta x \to 0} \left( \frac{\Delta u}{\Delta x} \right),$$

where $$\Delta u$$ is the total increment of $$u$$ caused by changes in all the variables, these increments being due to the change $$\Delta x$$ in the independent variable. In contradistinction to partial derivatives, $$\tfrac{du}{dt}, \tfrac{du}{dx}$$ are called total derivatives with respect to $$t$$ and $$x$$ respectively.