Perhaps a silly question, but suppose I have a differentiable function f(x,y), is the following statement true?
Serious calculus question

I don't know the answer but this is a good example of why calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other wellknown calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.

Perhaps a silly question, but suppose I have a differentiable function f(x,y), is the following statement true?
You get B in high school math.And only because your teacher was explicitly forbidden to hurt your feeling by grading anywhere lower.
What a bunch of pampered kids with elevated ungrounded selfesteem you are!

^ Yes, of course it should be:
Perhaps a silly question, but suppose I have a differentiable function f(x,y), is the following statement true?
\lim_{\varepsilon \rightarrow 0} \frac{f(x+\varepsilon,y+\varepsilon)f(x,y)}{\varepsilon}=\lim_{\varepsilon \rightarrow 0} \frac{f(x+\varepsilon,y)f(x,y)}{\varepsilon}+\lim_{\varepsilon \rightarrow 0} \frac{f(x,y+\varepsilon)f(x,y)}{\varepsilon}obvious. typo was obvious.

Yes, of course it should be:
[math] \lim_{\varepsilon \rightarrow 0} \frac{f(x+\varepsilon,y+\varepsilon)f(x,y)}{\varepsilon}=\lim_{\varepsilon \rightarrow 0} \frac{f(x+\varepsilon,y)f(x,y)}{\varepsilon}+\lim_{\varepsilon \rightarrow 0} \frac{f(x,y+\varepsilon)f(x,y)}{\varepsilon} [\math]
obvious. typo was obvious. 
OP's statement is correct. If the function is differentiable (at (x,y)), then one can replace a directional derivative (LHS) with the product of the gradient and the direction vector (RHS). It is true that the direction vector is often taken to have unit length, while here it is (1,1), so that the LHS is a directional derivative scaled by 2^0.5 but the scaling occurs on the RHS as well, and thus the equality still holds.

OP's statement is correct. If the function is differentiable (at (x,y)), then one can replace a directional derivative (LHS) with the product of the gradient and the direction vector (RHS). It is true that the direction vector is often taken to have unit length, while here it is (1,1), so that the LHS is a directional derivative scaled by 2^0.5 but the scaling occurs on the RHS as well, and thus the equality still holds.
Thank you! Finally a competent answer.