I am having a terrible hard time with the multivariable chain rule and its related stuff (I read my textbook many times, but it doesn't help that much because the explanations are very limited). I hope that someone can help me to withdraw from this darkness of confusion. 1) (Differentiation along curves) Find the rate of change of f(x,y) = 1 - x^2 - y^2 along the ellipse g(t) = (2cos(t),sin(t) ) at the point where t=pi/4. Using chain rule, we can get that the answer is d(f(g(t))/dt [evaluated at t=pi/4] =3. But what does this represent geometrically? I have no idea what is going on geometrically about differentiation along curves. Is the ellipse even lying on the surface f(x,y) = 1 - x^2 - y^2? If not, then why does it even make sense to talk about "the rate of change of f(x,y) = 1 - x^2 - y^2 along the ellipse g(t) = (2cos(t),sin(t) ) "? 2) From my textbook: "Suppose w= f(x,y,t,s) and that x and y are themselves functions of the independent variables t and s. [note: Let D=curly d, representing partial derivatives] If we write Dw/Dt = (Dw/Dx)(Dx/Dt) + (Dw/Dy)(Dy/Dt) + (Dw/Dt) using the chain rule, this is nonsense because the Dw/Dt on the left and right denote different things. We definitely cannot cancel out the two Dw/Dt." Now this is driving me crazy, when I use the chain rule, the above expression is what I get. How can this be wrong? I absolutely don't understand why the Dw/Dt would be denoting different things. And also, how can we cope with this problem? (i.e. what would be the correct way to write the expression?) 3) [note: Let D=curly d, representing partial derivatives] If, for example, w=f(x,y,z), are Dw/Dx and Df/Dx always always equal? My textbook seems to use Dw/Dx and Df/Dx quite interchangably in most cases, but I am not sure whether they are ALWAYS equal. The variable-dependence thing is just driving me crazy... Any help is greatly appreciated!