Question

Compute a second-order Taylor series expansion around point (a,b) = (0,0) of a function
f(x,y)=e^x*log(1+y)

Answer 1

http://en.wikipedia.org/wiki/Taylor_series
The example at the end of the page is exactly your problem

Answer 2

a taylor series of multivariables correlates to the partial derivatives

Answer 3

Fx is the term for x

Fxy is the term for xy

and so on

Answer 4

Another way to this is to write
\[e^x =1+x+\frac{x^2}{2}+\frac{x^3}{6
}+O\left(x^4\right)\\
\ln( 1+ y)=y-\frac{y^2}{2}+\frac{y^3}{3}+
O\left(y^4\right)\\

e^x \ln(1+y)=\left ( 1+x+\frac{x^2}{2}+\frac{x^3}{6
}+O\left(x^4\right) \right) \left ( y-\frac{y^2}{2}+\frac{y^3}{3}+
O\left(y^4\right) \right)


\]
Multiply out and you can find some few terms of your series expansion.