Primer of Modern Analysis pp 278-288 | Cite as

# Higher Derivatives

Chapter

## Abstract

The partial derivatives of a function

*f:***R**^{m}→**R**^{n}are again functions from**R**^{m}to**R**^{n}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.

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## Copyright information

© Springer Science+Business Media New York 1983