SixThousandMono Good question. The relationship between the two inputs of the function may not be known, or you might want the result of whatever you're doing to apply for many different possible relationships between the two inputs. The relationship may also be complex enough where eliminating one of the variables would create a more algebraically complex expression, in which case it's up to you whether you'd rather deal with a somewhat simple two-variable function or a one-variable mess. The final reason I can think of to not eliminate a variable is that each variable might have some physical meaning. In thermodynamics, pressure and volume are often related in a known way, but it may still be advantageous to write a function with both P and V variables so it's easier to wrap your mind around what the equation is telling you. Hope that helps!

Wtf is this question

Agreed.

Good question. We're assuming y depends on x, just in case it does. This is a safe assumption, because if it's false, the dy/dx in the last term would just be zero anyway. The point of the full derivative is to take into account the fact that your input variables might not be totally independent. You might not run into this in a regular multivariable calc class, since your input variables are usually just coordinates in cartesian space. But if you're in a different coordinate system or working with some thermodynamic functions where everything is tied together, this is useful.