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

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

where and are functions of a third variable .

Let take on the increment , and let , , be the corresponding increments of , , respectively. Then the quantity

is called the total increment of .

Adding and subtracting in the second member,

(A)

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,

(B)
[, and since varies while remains constant, we get the partial derivative with respect to .]

For the second difference we get

(C)
[ and since varies while remains constant, we get the partial derivative with respect to .]

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

(D)

where and are positive proper fractions. Dividing (D) by ,

(E)

Now let approach zero as a limit, then

(F)
[Since and converge to zero with , we get and and being assumed continuous.]

Replacing by in (F), we get the total derivative

(51)