If $\displaystyle z \:= \:f(x,y) \:= \:x^3 + y^2 - 8xy + 2x^3y^2 + 11$, find

a) the partial derivative of $\displaystyle f$ with respect to $\displaystyle y$

b) the partial of $\displaystyle f_y$ with respect to $\displaystyle y$

The method I am using is to treat all x's as constants and to take the derivative of all y's.

For a) I get: $\displaystyle f_y \:= \:0 + (y^2)' - (8x)(y)' + (2x^3)(y^2)' + 0$

. . $\displaystyle = \:2y - 8x(1) + (2x^3)(2y) \:= \:2y - 8x + 4x^3y$ . . . which matches the book's answer.

For b) I use the same approach.

If $\displaystyle f_y \:=\: 2y - 8x + 4x^3y$, then $\displaystyle f_{yy}$ should be:

. . $\displaystyle f_{yy} \:= \:2(y)' - (0) + (4x^3)(y)' \:= \:2 (1) + (4x^3)(1)\:=\:2 + 4x^3$

* . . . Correct!*
The book, however, gives the answer as: $\displaystyle 2 + 12x^2y$

** ??**
Am I missing something?

. . . Not a thing!