Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 7]_Higher-Order Derivatives and Differentials_197
1889. Find the steepest slope of the surface
z = x' + 4y*
at the point (2, 1, 8).
1890. Construct a vector field of the gradient of the following
functions:
a) z = je-f y: c) z = x* + y z
\
Sec. 7. Higher-Order Derivatives and Differentials
1. Higher-order partial derivatives. The second partial derivatives of a
function z /(*, y) are the partial derivatives of its first partial derivatives.
For second derivatives we use the notations
d (dz\ 3 2 z
Derivatives of order higher than second are similarly defined and denoted.
If the partial derivatives to be evaluated are continuous, then the result
of repeated differentiation is independent of the order in which the differentiation
is performed.
Example 1. Find the second partial derivatives of the function
z~ arc tan .
y
Solution. First lind the first partial derivatives:
dz
== _J_ y
m j__
dz^
dy~~ ,
Now differentiate a second time:
^L = . d f y \- 2 *y
d z z .
dxdy
We note that the so- so-called "mixed" partial derivative may
different way, namely: el
d /
dxdy~~dydx
dx(
1_
J
2
be found in a
y*