Here is the proof of Noether's Theorem given in Peskin's and Schroeder's book on QFT:
What I don't understand is why he writes that $\Delta \mathcal{L} = \frac{\partial\mathcal{L}}{\partial\phi}\Delta\phi + \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\Delta(\partial_\mu\phi)$. I do not see why these $\Delta$ should be small enough so that we could do something like $\Delta f=f'\Delta x$.