Taylor series of x^y

[SilverKing]

Hi everyone,

I've the following problem:

"The following formula represents Taylor series of the function F at Mo(Xo,Yo)
F(X,Y)=F(Xo,Yo) + 1/1! dF(Xo,Yo) + 1/2! d^2F(Xo,Yo) + + 1/3! d^2F(Xo,Yo) + ....

Find the first three nonzero limits of the Taylor series for the function F(X,Y)=x^y at Mo(1,1)"

I've tried to solve the problem by finding the first derivatives of x^y

First order: dF=dF/dx dx + dF/dy dy
Second order: d^2F=d^2F/dx^2 (dx^2) + 2d^2F/dxdy (dxdy) + d^2F/dy^2 (dy^2)
Third Order: d^3F=d^3F/dx^3 (dx^3) + 3d^3F/dxdy (dx^2 dy) + 3d^3F/dxdy (dx dy^2) + d^3F/dy^3 (dy^3)

First order would be: (y-1) x^y-1 (dx) + x^y ln x (dy)

at (1,1)
(1-1) 1^1-1 dx + 1^1 ln 1 (dy)
1 dx + 0 (dy)
so dF(1,1)=dx

and so on


are steps correct?