I am doing a chapter on bivariate calculus and so far have done about 30 problems without any mistakes. Yet I've come across one where I can't match the author's answer. Given that I've found a few typos before, I'd like to know if I'm right or wrong, and if wrong, why.

The relevant part of the problem asks:

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

a) the partial derivative of f with respect to y, and

b) the partial of f [subscript] y (i.e. the 2nd derivative) with respect to y.

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

Thus, when y' means "derivative of y", and when

f(x,y) = x^3 + y^2 - 8xy + (2x^3)(y^2) + 11,

For a) I get

(0) + (y^2)' - (8x)(y)' + (2x^3)(y^2)' + (0)

= 2y - 8x(1) + (2x^3)(2y)

= 2y - 8x + (4x^3)(y),

which matches the book's answer.

For b) I use the same approach.

If the partial derivative of f with respect to y

= 2y - 8x + (4x^3)(y),

then

the partial of f [subscript] y with respect to y should be

= 2(y)' - (0) + (4x^3)(y)'

= 2 (1) + (4x^3)(1)

= 2 + 4x^3.

The book, however, gives the answer as 2 + (12x^2)(y), without showing any work.

Am I missing something?