Dear Sir
In calculus, we often mention to the students that if F(x,y) = 0, then we can differentiate both sides and still get an equality. The problem is that we can't perform the same operation on F(x) = 0, say x = 0, otherwise 1 = 0, which is absurd. What is the reason?
Jacob
(Grade 12 question)
Nice Question.
One way to sort this out to is to think of the picture, in the plane.
The derivative is a measure of the amount that y changes when you change x. F(x,y) = 0 is (typically) a curve - and curves have room to 'change x', have tangents, etc. In fact, curves were the basis of calculus from Newton and Leibnitz up to around 1810. Only then did mathematicians shift to the idea of a 'function', of limits, etc. as we now teach them.
F(x)= 0 is (typically) a set of isolated points, with no use of y. If you insist on inserting y, you find that the graph is typically a set of vertical lines. Either way, it is not a good candidate for 'changing x'.
For example if F(x) = x2 -x - 2 = (x-2)(x+1) = 0 then the graph is all points (x,y) in the plane that satisfy F(x) = 0. This graph is two vertical lines at x = -1 and x = 2.
Of course F(x,y) may not actually have y occurring in the equation - it may be the second case in disguise. The second case may not have any x - it may be 0=0 (the entire plane) or 1=0 no points at all. Just the act of writing something with a certain format does not guarantee that everything works fine. We get conditioned to 'typical patterns'. Every once and a while we need to stop and look to see if our example is 'typical'.
Walter