## The Calculus Primer (2011)

### Part X. Partial Differentiation

### Chapter 37. SIGNIFICANCE OF PARTIAL AND TOTAL DERIVATIVES

**10—6.Partial Differentials and Partial Derivatives.** Some notion of the significance of partial differentials will be obtained from the following examples. Consider the changes in the area of a rectangle caused by variation in the lengths of the sides *separately,* and when varying *simultaneously.* Let the rectangle *ABCD* have a variable base *x* and a variable altitude *y*; let its area equal *z* = *xy.* Now if we consider *x* constant while *y* increases by an increment *BP = dy,* the corresponding change in *z* is *PQCB* = *dy*; if we consider *y* constant while *x* increases by *DS* = *dx,* the corresponding change in *z* is *CBSD* = *dx*; and the total differential of *z* is given by

Moreover, from the equation *z* = *xy,* we see that

It should also be carefully noted that the total increment Δ*z* due to increments *dx* and *dy* is given by

Δ*z* = *x dy + y dx* + *dx dy*;

the small rectangle *RQ,* whose area equals *dx dy,* represents the difference between Δ*z* and *dz.* Thus the total increment and the total differential of a function of two or more variables are not, in general, equal.

EXAMPLE. The total area *A* of a right circular cone is given by *A* = *πrs + πr*^{2}*,* where *s* = slant height and *r* = radius of base. Find the rate of change of *A* with respect to r when s remains constant; the rate of change of *A* with respect to *s* when r is constant.

*Solution.*

*A* = π*rs + πr*^{2};

**10—7.The Total Derivative.** As we have already learned, the total differential of *u* = *f*(*x*,*y*) is

if now *x* and *y,* and therefore *u*,are functions of another variable *t*, we may divide (1) by *dt*:

This, as we have seen, is the total derivative of *u* with respect to *t*; here and may be regarded as derivatives which represent the rates of change of *x* and *y*, respectively, with respect to *t*. In practical applications, the variable *t*often represents time.

EXAMPLE 1.The base of a rectangle equals 8 inches, and its altitude is 4 inches. At a certain instant the base is increasing at the rate of 3 in./sec, and the altitude at the rate of 2 in./sec. At what rate is the area changing at the same instant?

*Solution.* Let *x* = base, *y* = altitude, *u* = area.

By equation (2) above:

But *x* = 8 in., *y* = 4 in., = 3 in./sec, = 2 in./sec.;

hence = (4) (3) + (8) (2) = 28 sq. in./sec.

EXAMPLE 2.Assume that the formula for a gas is *pv* = *kT,* where *p* = pressure in lb./sq. unit, *v* = the volume in corresponding cubic units, *T* = absolute temperature, and *k* = a constant. If *k* = 60, and at a certain instant *v* = 12 cu. ft. and *T* = 200°, find *p* at this instant. If at the same instant *v* is increasing at the rate of 0.5 cu. ft./min., and *T* is increasing at 0.6 degree/min., at what rate is *p* changing at that instant?

*Solution.* Substituting *k* = 60, *v* = 12, *T* = 200 in the formula *pv* = *kT*, we find *p* = 1000. At the instant in question,

Now, since , we note that

In other words, the pressure is *decreasing* at the rate of 38.7 lb./sq. ft./minute.