If y = f(x) , then we have the differential
dy = f'(x) dx
where f'(x) is the derivative of y with respect to x. Thus, we have (for the case when dx not equal to zero),
dy/dx = f'(x)
so that the derivative is a ratio of differentials. Note that we don't have to worry about dx equalling zero because we only consider the limiting case as dx goes toward zero.
These do indeed follow canceling rules as well (this is a touch mathy). Suppose we have y = f(x), where f is continuous then x = f-1(y) (the inverse of f, which is also continuous). Let's let g = f-1. Consider the slope of the secant line between a point (y,g(y)) and another point (b,g(b)). The slope is given by:
g(y) - g(b)
y - b
g(y) is some x and we will have g(b) = a. We then take the limit as y goes to b, which gives g'(b) Since f and g are continuous we can write the above as
x - a
f(x) - f(a)
In the limit as x goes to a. We have used the fact that g and f are inverses of each other. We then use the limit laws so that:
x - a 1
----------- = -----------------
f(x) - f(a) (f(x)-f(a))/(x-a)
In the limit as x goes to a
And thus, g'(b)f(a) = 1, or equivalently, we can state (since f(a) = b and we can easly extend these to any such x and y)
-- -- = 1
The difference above is that those derivatives are partial derivatives (that is, derivatives of a functions of more than one variable while holding one variable constant in which we cannot treat the derivatives as ratios of differentials.
Of course this is in the case where f'(x) does not equal zero.