Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/219

 We shall next consider a function of two variables, both of which depend on a single independent variable. Consider the function

where $$x$$ and $$y$$ are functions of a third variable $$t$$.

Let $$t$$ take on the increment $$\Delta t$$, and let $$\Delta x $$, $$\Delta y$$, $$\Delta u$$ be the corresponding increments of $$x$$, $$y$$, $$u$$ respectively. Then the quantity

is called the total increment of $$u$$.

Adding and subtracting $$f( x, y + \Delta y)$$ in the second member,

Applying the Theorem of Mean Value (46), §106, to each of the two differences on the right-hand side of (A), we get, for the first difference,

For the second difference we get

Substituting (B) and (C) in (A) gives

where $$\theta_1$$ and $$\theta_2$$ are positive proper fractions. Dividing (D) by $$\Delta t$$,

Now let $$\Delta t$$ approach zero as a limit, then

Replacing $$f(x, y)$$ by $$u$$ in (F), we get the total derivative