## The Calculus Primer (2011)

### Part X. Partial Differentiation

### Chapter 36. THE TOTAL DERIVATIVE

**10—4.Tolal Differential.** We have already learned how to find the derivative of a function of a function; thus, if *y* = *f*(*u*), and *u* = *F*(*x*), then

In a similar way, it may be shown that if *z* = *f*(*x,y*), and *x* = *F*_{1} (*t*) and *y* = *F*_{2}(*t*) then

Equation [1] gives the *total derivative* of *z* with respect to *t.* In the same way, if *u = f*(*x*,*y*,*z*), and *x*, *y*, and *z* are functions of *t*, then

It should be understood that the meanings of and are definitely different. Thus the partial derivative supposes that *only the particular* *variable x varies;* on the other hand, is the limit of , where Δ*u* is the *total increment* in *u* brought about by *changes in all the variables due* *to an increment in the independent variable x.*

If we multiply equations [1] and [2], respectively, by *dt,* we obtain

The expression *du* is called the *total differential of u*, or the “complete differential.”

The expressions *dx*, *dy*, etc. are called *partial differentials.*

**10—5.Implicit Functions.** The expression *f*(*x*,*y*) = 0 represents an equation in *x* and *y* where all the terms have been transposed to one side of the equation. In other words, *y* is an implicit function of *x* (or *x* is an implicit function of *y*). Now, let us set *z* = *f*(*x,y*)*,* and find the total differential of *z;* from [3], §10—4, we have:

But by hypothesis, since *z* = *f*(*x,y*) = 0, then for all values of *x, z* = 0; hence *dz* = 0. Therefore, if ≠ 0:

or, dividing (2) through by *dx* and by , we get:

EXAMPLE. If *f*(*x,y*) = *x*^{2} + *y*^{3} + *xy* = 0, find by using partial derivatives.

*Solution.*

This result agrees, of course, with the value of found by the previously learned method; thus, differentiating “directly”:

The relation above is quite general. Thus if

*u* = *F*(*x*,*y*,*z*) = 0,

then, by §10—4, equation [4]:

Analogous to the derivation in the earlier part of the present section, we remember that *z* is a function of *x* and *y,* so that

Substituting the value of *dz* from (2) in equation (1), and remembering that *du =* 0, we obtain:

or, by factoring:

Since *z* is a function of *x* and *y,* the variables *x* and *y* are independent variables, and so we may assign to the increments *dx* and *dy* such values as we please.

Let us first set *dy* = 0, ≠ 0; then, from (3), we get:

Next, we set *dx* = 0, *dy* ≠ 0; then, from (3) we get: