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230 • Chapter 6: Examples from Calculus> fy(0,0) := limit( ( f(0,k) - f(0,0) )/k, k=0 );fy(0, 0) := 0Here is a mixed second derivative of f.> fxy := D[1,2](f);fxy := (x, y) → x2 − y 2x 2 + y 2 + 2 x 2x 2 + y 2 − 2 x2 (x 2 − y 2 )(x 2 + y 2 ) 2− 2y 2x 2 + y 2 − 2 y2 (x 2 − y 2 )(x 2 + y 2 ) 2 + 8 x2 y 2 (x 2 − y 2 )(x 2 + y 2 ) 3Again, the formula does not hold at (0, 0).> fxy(0,0);Error, (in fxy) numeric exception: division by zeroThe limit definition is> Limit( ( fx(0,k) - fx(0,0) )/k, k=0 );lim − 1k→0> fxy(0,0) := value( % );fxy(0, 0) := −1The other mixed second derivative is> fyx := D[2,1](f);fyx := (x, y) → x2 − y 2x 2 + y 2 + 2 x 2x 2 + y 2 − 2 x2 (x 2 − y 2 )(x 2 + y 2 ) 2− 2y 2x 2 + y 2 − 2 y2 (x 2 − y 2 )(x 2 + y 2 ) 2 + 8 x2 y 2 (x 2 − y 2 )(x 2 + y 2 ) 3At (0, 0), you need to use the limit definition.> Limit( ( fy(h, 0) - fy(0,0) )/h, h=0 );