To prove that the field of meromorphic functions on $X(1)$ is generated by the Klein j-invariant, we need to show that the derivative of $j(\tau)$ does not vanish. (It only has simple zeroes).
But I don't know how to prove this. In Fred Diamond and Jerry Shurman's book "A First Course in Modular Forms", page 73, they just say that $j'(\tau)$ does not vanish.