What is meaning of dy/dx?

Remember back to basic Algebra I, when the slope of a line was given by m=?y/?x? And, what is a derivative, but the slope of a (tangent) line? Indeed, if you look back at the limit definition of the derivative, it's really nothing more than the limit of ?y/?x as ?x?0. Thus, you can think of the "d" in "dy" and "dx" as a very tiny delta. (Indeed, Leibniz' notion of the derivative involved infinitessimal ?'s; unfortunately, it would take a couple centuries before a formal notion of "infinitessimal" could be developed to rescue Leibniz!) So, just like in ?y/?x, the "d" means "change." And, just like in ?y/?x, it really doesn't make much sense to talk about ?y without ?x, or vice versa! (That is, they represent a RELATIVE change: the change in y relative to a certain change in x.) However, in differential equations, we can split the dy from the dx on opposite sides of the equality, because we're still maintaining the relationship between dy and dx. For example, if dy/dx = y²sinx, then the equation dy/y² = sinxdx really tells how dy and dx are related to each other! read less