# Help with a second partial derivative, please

• May 29th 2006, 06:14 AM
lingyai
Help with a second partial derivative, please
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),

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?
• May 29th 2006, 08:28 AM
Soroban
Hello, lingyai!

Quote:

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!
The \$\displaystyle 12x^2y\$ looks like they differentiated with respect to x . . . but even that is wrong.

If \$\displaystyle f_y\:=\:2y - 8x + 4x^3y\$, then: \$\displaystyle f_{yx}\:=\:-8 + 12x^2y\$

Since \$\displaystyle f_x\:=\:3x^2 - 8y + 6x^2y^2\$, then \$\displaystyle f_{xx} \:=\:6x + 12xy^2\$

Their answer doesn't match anything in the problem!
. . It must be just one more typo on their part.
• May 29th 2006, 08:35 AM
lingyai
Thanks very much for that feedback. Having only started with multivariate calculus today, I was inclined to assume I'd missed something. Not being able to figure it out was troubling.

I'm all in favor of students learning from "spot the deliberate mistake" type answers, but typos in what is supposed to be a self-contained, self-teaching text are very annoying!

Thanks again,

Ken (lingyai)