Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/222

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are called partial differentials. These partial differentials are sometimes denoted by dxu, dyu, dzu, so that (56) is also written

Illustrative Example 1. Given , find .

Solution.



Substituting in (55),

Ans.


Illustrative Example 2. The base and altitude of a rectangle are 5 and 4 inches respectively. At a certain instant they are increasing continuously at the rate of 2 inches and 1 inch per second respectively. At what rate is the area of the rectangle increasing at that instant?

Solution. Let base, altitude; then area,

Substituting in (51),

(A)

But in., in., in. per sec., in. per sec.

sq. in. per sec. sq. in. per sec. Ans.

NOTE. Considering as an infinitesimal increment of area due to the infinitesimal increments and , is evidently the sum of two thin strips added on to the two sides. For, in (multiplying (A) by ),

= area of vertical strip, and
= area of horizontal strip.

But the total increment due to the increments and is evidently

Hence the small rectangle in the upper right-hand corner () is evidently the difference between and . This figure illustrates the fact that the total increment and the total differential of a function of several variables are not in general equal.

127. Differentiation of implicit functions. The equation

(A)

defines either or as an implicit function of the other.[1] It represents any equation containing and when all its terms have been transposed to the first member. Let

(B)  
then (53), §125
  1. We assume that a small change in the value of causes only a small change in the value of .