Taylor series expansion of $\sec(x +y^2)$ We have $f(x,y) = \sec(x+y^2)$ 
I want to find the first two non-zero terms of $f$ at $(0,0)$ starting by 


*

*Taking the first few terms of $\cos x$ centered at zero, $1 - \frac{x^2}{2!} $

*Using this to get the first few terms of $\sec x$ centered at zero, $\frac{1}{1 - \frac{x^2}{2!}} =  1+\frac{x^2}{2} + \cdots = 1+\frac{x^2}{2} $ (see comment). 

*I don't know how to use these to find the first two non-zero terms of $f$, which we are supposed to do. 
 A: You already have the first few terms of the Taylor expansion of $\sec w$ about $w=0$ (yes, the choice of variable is deliberate).
Everywhere that you see a $w$ in that expansion, write $x+y^2$. Now it depends how many terms you want. But from the wording of the question, it looks as if expanding the $w^2$ term will be enough, and you won't even need the "$y^4$" term.
Alternately, and as a check, look up the formula for the Taylor expansion of a function $f(x,y)$ of two variables. You will need to evaluate some higher partial derivatives, since the first partials are $0$ at $(0,0)$.  From the wording of the question, that is not what you are being asked to do. The question is asking you to "recycle" a known expansion to obtain a new one.
Here is a simpler example to illustrate the idea. Suppose you want the Taylor expansion of $\sin(xy^2)$.  We could take partial derivatives, but that's doing things the hard way. Just find the Taylor expansion of $\sin w$, and everywhere that you see $w$, substitute  $xy^2$.    
