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Homework Statement
Hello, I'm trying to grasp the definition of a derivative in several variables, that is, to say if it's differentiable at a point.
My book tells me that a function of two variables if differntiable if:
[tex]f(a+h,b+k)f(a,b) = A_1h+A_2k+\sqrt{h^2+k^2}\rho(h,k)[/tex]
And if [tex] \rho[/tex] goes to zero as (h,k) > 0. First of all, how did one "come up" with this? It seems a bit arbitrary to me, which I am sure it is not.
Apart from that, I'm trying to do show that a derivative exists at a point:
[tex]f(x,y) = sin(x+y) [/tex] at (1,1)
Homework Equations

The Attempt at a Solution
To show that the derivative exists:
[tex]f(1+h,1+k)f(h,k) = \sin(2+(h+k)) \sin(2) = 0*h+0*k+\sqrt{h^2+k^2}\rho(h,k)[/tex]
and:
[tex] \rho(h,k) = \frac{\sin(2+(h+k))}{\sqrt{h^2+k^2}} \text{ if } (h,k) \neq 0 \text{ and } \rho(h,k) = 0 \text{ if } (h,k) = 0 [/tex]
Then I guess I'm supposed to prove that the limit goes to zero, but how do I do it in this case?