Part 4: Higher Derivatives

Higher partial derivatives are defined similarly. For example, the third derivative of f with respect to x is the partial derivative with respect to x of the second derivative f_xx. That is,

fxxx( x,y) =  ¶ ¶x fxx(x,y)

Similarly, f_xxy is defined

fxxy( x,y) =  ¶ ¶y  fxx(x,y)

In operator notation, the partial derivative of f for m times with respect to x and n times with respect to y is denoted by

¶m+nf ¶xm ¶yn

The m+n partial derivatives of f( x,y) are then defined in terms of the previous partial derivatives as

¶m+nf ¶xm ¶yn =  ¶ ¶x  ¶ ¶y æ è  ¶m+n-2f ¶xm-1¶yn-1 ö ø

when f( x,y) and its partial derivatives are continuous on a region through the m+n order.       

EXAMPLE 8    Find f_xxyy if f( x,y) = x⁴y⁴.       

Solution: It is easy to show that f_xx = 12x²y⁴. Thus,

fxxy =  ¶ ¶y fxx =  ¶ ¶y 12x2y4 = 48x2y3

and similarly,

fxxyy =  ¶ ¶y fxxy =  ¶ ¶y 48x2y3 = 144x2y2

Moreover, we usually assume that f is sufficiently smooth at all points where partial derivatives are defined so that mixed partials are independent of the order of differentiation. Indeed, notice that if f(x,y) = x⁴y⁴, then

fxyx =  ¶ ¶x fxy =  ¶ ¶x 16x3y3 = 48x2y3

which is the same as f_xxy in example 1. In addition, f_xyxy = 144x²y² = f_xxyy.