What are you having troubles with? Do you know how to differentiate partially? If it is you just differentiate in respect to holding constant...same concept for . Try it out and report back with problems?

What is meant by ? My book uses the notation to denote the directional derivative...which doesnt appear to be the case since you did not supply a unit position vector...so is this the so called "total derivative"?2.

Let f:R2 ->R be defined by f(x,y) = x^2 - 3y^2 and G:R2 ->R2 be defined by G(s,t) = {st,s+(t^2)}. Calculate Df,Dg and eventually D(f o G)(1,2).

and one that I can't remember the method for (not as important)...

Convert to polar coordinates. Note that when you convert since approaches zero from all paths there is no need to consider it3.

f(x,y) = xy / sqrt(x^2 + y^2), if (x,y) does not = (0,0), f(0,0) = 0

Prove the f is continuous at (0,0)

so

Which clearly is independent upon

To illustrate this point further note that

which clearly tends to zero as

So we can conclude that

thus continuous.