Higher Derivatives

Abstract

The partial derivatives of a function f:R^m → Rⁿ are again functions from R^m to Rⁿ which may have partial derivatives of their own. If so, the latter are called the second partial derivatives:

$$\frac{{{\partial ^2}f\left( a \right)}}{{\partial {x_i}\partial {x_j}}} = {D_{ij}}f\left( a \right) = {D_i}\left( {{D_j}f} \right)\left( a \right)$$

(1)

The formula says that to get D _ij f(a), you take first Dif, which must exist on a neighborhood of a, and differentiate it with respect to xi. It appears that this would be quite different from D _ji f (a), which is formed by differentiating first with respect to x _i and then with respect to x _j .In fact, the two are usually the same.