| dnƒ = ( dx | ∂ | + dy | ∂ | ) | n | ƒ |
| ∂x | ∂y |
| = (dx)n | ∂nƒ | + n(dx)n−1 dy | ∂nƒ | + ... |
| ∂xn | ∂xn−1 ∂y |
in which the expression (dx·∂/∂x + dy·∂/∂y)n is developed by the binomial theorem in the same way as if dx·∂/∂x and dy·∂/∂y were numbers, and (∂/∂x)r·(∂/∂y)n−r ƒ is replaced by ∂nƒ/∂xr ∂yn−r. When there are more than two variables the multinomial theorem must be used instead of the binomial theorem.
The problem of forming the second and higher differential coefficients of implicit functions can be solved at once by means of partial differential coefficients, for example, if ƒ(x, y) = 0 is the equation defining y as a function of x, we have
| d2y | = ( | ∂ƒ | ) | −3 | { ( | ∂ƒ | )2 | ∂2ƒ | − 2 | ∂ƒ | · | ∂ƒ | · | ∂2ƒ | + ( | ∂ƒ | )2 | ∂2ƒ | }. |
| dx2 | ∂y | ∂y | ∂x2 | ∂x | ∂y | ∂x∂y | ∂x | ∂y2 |
The differential expression Xdx + Ydy, in which both X and Y are functions of the two variables x and y, is a total differential if there exists a function ƒ of x and y which is such that
∂ƒ/∂x = X, ∂ƒ/∂y = Y.
When this is the case we have the relation
∂Y/∂x = ∂X/∂y.
(ii.)