| lim.h=0, k=0 | R | = 0 and lim.h=0, k=0 | R | = 0. |
| h | k |
In accordance with the notation of differentials this equation gives
| dƒ = | ∂f | dx + | ∂y | dy. |
| ∂x | ∂y |
Just as in the case of functions of one variable, dx and dy are arbitrary finite differences, and dƒ is not the difference of two values of ƒ, but is so much of this difference as need be retained for the purpose of forming differential coefficients.
The theorem of the total differential is immediately applicable to the differentiation of implicit functions. When y is a function of x which is given by an equation of the form ƒ(x, y) = 0, and it is either impossible or inconvenient to solve this equation so as to express y as an explicit function of x, the differential coefficient dy/dx can be formed without solving the equation. We have at once
| dy | = − | ∂ƒ | / | ∂ƒ | . |
| dx | ∂x | ∂y |
This rule was known, in all essentials, to Fermat and de Sluse before the invention of the algorithm, of the differential calculus.
An important theorem, first proved by Euler, is immediately deducible from the theorem of the total differential. If ƒ(x, y) is a homogeneous function of degree n then
| x | ∂f | + y | ∂f | = nƒ(x, y). |
| ∂x | ∂y |
The theorem is applicable to functions of any number of variables and is generally known as Euler’s theorem of homogeneous functions.