# Showing a multivariable function is differentiable at a point

So I've been asked a question: "Show the function $$f(x,y) = x^2 - xy + 3y^2$$ is differentiable at the point $$(1,1)$$ using the definition of differentiability." The question also mentions that it wants us to indicate the linear and little-o terms, and prove the little-o terms are in fact little-o terms.

I've been given the answer already (see below) and understand most of what is going on but I'm not sure how we go from line 3 to line 4 (line 4 being the one that starts with $$f(1,1)$$).

I understand what happens after line 4, it's just the jump from line 3 to line 4 that I don't understand, where to these terms ($$o(h)$$, $$A h$$, and $$f(1,1)$$) come from?

From line 3 to line 4:

$$f(1,1)=3$$

$$h=(h_1,h_2)$$ and $$Ah=h_1+5h_2$$

$$h_1^2-h_1h_2+h_2^2=o(h)$$

• ok, i get the f(1,1) part, he just subbed x=1 and y=1 into the original equation. For A, did he find the value of A just by seeing the values of the constants on h1 and h2 were 1 and 5? How did o(h) appear, is that just a general formula for little-o? Jan 28, 2021 at 15:18
• @Maximus: $h_1+5h_2$ is the dot product of two vectors, $(1,5)$ and $(h_1,h_2)$.
– user9464
Jan 28, 2021 at 15:20
• @Maximus $h_1^2-h_1h_2+h_2^2=o(h)$ by definition of little o. Note particularly that $h=(h_1,h_2)$.
– user9464
Jan 28, 2021 at 15:21
• Ohhhh, ok thanks a lot, was actually pretty simple I just didn't look at it the right way. Jan 28, 2021 at 15:25
• @Maximus You are welcome. :-)
– user9464
Jan 28, 2021 at 15:26