Given a funtion y(x), then d(y(x))(h) (or dy for short) is the differential of y(x) which is defined as y'(x)*h where y'(x) is the derivative of y(x) with respect to x and h is a new variable.
The same goes for dx where x = i(x) is the identity function with respect to x. So given x' = 1 we have dx = h. So dy(x)/dx is just another way to write y'(x).