# Higher Derivatives

• Kennan T. Smith
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

The partial derivatives of a function f:RmRn are again functions from Rm 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 xi and then with respect to x j .In fact, the two are usually the same.

## Keywords

Partial Derivative Local Maximum Linear Transformation Smooth Manifold High Derivative
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.