Chapter 4: Partial Differentiation
Section 4.2: Higher-Order Partial Derivatives
Example 4.2.5
If f={x y x2−y2x2+y2x,y≠0,00x,y=0,0 and a,b=0,0, obtain all second partial derivatives, both at x,y and at a,b.
Solution
The first partial derivatives are given in Table 4.1.5(b), reproduced here as Table 4.2.5(a).
fx={yx4+4x2y2−y4x2+y22x,y≠0,00x,y=0,0
fy={xx4−4x2y2−y4x2+y22x,y≠0,00x,y=0,0
Table 4.2.5(a) First partial derivatives of fx,y.
Table 4.2.5(b) states the second-order partial derivatives. Note in particular that the mixed partial derivatives are not equal. That is the point of this example, namely, that without some qualification on a multivariate function, the mixed partials need not be equal.
fxx={−4xy3x2−3y2x2+y23x,y≠0,00x,y=0,0
fxy={x6+9x4y2−9x2y4−y6x2+y23x,y≠0,0−1x,y=0,0
fyy={−4yx33x2−y2x2+y23x,y≠0,00x,y=0,0
fyx={x6+9x4y2−9x2y4−y6x2+y23x,y≠0,01x,y=0,0
Table 4.2.5(b) Second-order partial derivatives of f
Table 4.2.5(c) lists the limits that define the values of the second-order partial derivatives at 0,0.
fxx0,0=limh→0fxh,0−fx0,0h=limh→00h=0
fyy0,0=limk→0fy0,k−fy0,0k=limk→00k=0
fxy0,0=limk→0fx0,k−fx0,0k=limk→0−kk= −1
fyx0,0=limh→0fyh,0−fy0,0h=limh→0hh=1
Table 4.2.5(c) Second-order partial derivatives at 0,0
Figure 4.2.5(a) is a graph of the surface corresponding to fx,y, whereas Figure 4.2.5(b) is a graph of the surface corresponding to either of the mixed second-order partial derivatives for x,y≠0,0. While the first surface seems smooth enough, the second reveals the discontinuity at the origin for the mixed second-order partial derivatives.
Figure 4.2.5(a) Surface for fx,y
Figure 4.2.5(b) Mixed partial, x,y≠0,0
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