Question

What is the difference between d/dx, dy/dx, and y'

Answer 1

d/dx is the derivative of any particular function with respect to x, as in: d/dx(x^2 + 1)

dy/dx is the derivative of the function y with respect to x because the function y is a function of x, i.e. y(x).
So, dy/dx = dy(x)/dx

y' is shorthand notation for dy/dx, i.e. y'=dy/dx

If you go to the wikipedia article for derivative and look in the "Notations for differentiation" section, it explains the differences in the notation.