Maclaurin series for multivariable

[manenbu]

Homework Statement

I got a few functions I need to expand to series using Maclaurin forumlas.

Homework Equations

http://mathworld.wolfram.com/MaclaurinSeries.html

The Attempt at a Solution

So here are the ones I managed to do:
f= \sqrt{1-x^2-y^2}
writing it in another form:
f= (1+(-x^2-y^2)^{\frac{1}{2}}
Then I use:
(1+x)^m = 1 + mx + ..
as a function of one variable where x = -x^2-y^2 and I get the correct answer.
same goes for
z=\frac{1}{1-x+2y}
and
p=\ln(1+x+y).
Basically - I found that whenever there is no multiplication involved, I can just treat the two variables as one big variable and it works (according to my given answers).
The problem comes when I got stuff like this:
g=\frac{\cos{x}}{\cos{y}}
or
v=e^{x}\cos{y}.
Expanding each part and then dividing or multiplying (as you would you do if it was a true single var function) doesn't work. Expanding with taylor series from the start works - but the point is to use the maclaurin series.
So where did I go wrong?

 

[HallsofIvy]

?? The MacLaurin series is a Taylor's series, just evaluated at x= 0.

 

[manenbu]

I know, but the MacLaurin series are given equations so you will not have to differentiate all over again. For example, for functions in the form of e^x or siny. You just plug it in the formula.
My question is - can it be done for functions like f(x,y) = e^xsiny? If yes - how? This way the differentiation process can be avoided.