Derivation of (2.45) in Peskin and Schroeder I'm having trouble understanding the step 
$$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int d^{3}y ~(-i\delta^{(3)}(\vec{x}-\vec{y})(-\nabla^{2} +m^{2})\phi (\vec{y},t)) $$
I've tried using the relations $$[\phi (\vec{x},t), \pi (\vec{y},t)] = i\delta^{(3)}(\vec{x}-\vec{y})$$ and $$[A,BC] = [A,B]C + B[A,C], $$ but run into $$[\pi (\vec{x},t), (-\nabla^{2} +m^{2})\phi (\vec{y},t)] ,$$ which I don't know how to evaluate.
Any help would be appreciated.
 A: This is really straight forward, once you get used to the notation. (Don't you hate it when people say that?)
$$[\pi (\vec{x},t), (-\nabla^{2} +m^{2})\phi (\vec{y},t)] ,$$
Here you need to remember that $\nabla^2$ acts on the $\phi(\vec{y},t)$ only, so $\pi$ can pass right through this wave operator. Now when you evaluate the commutator you'll end up with something like $\phi (\vec{y},t)(-\nabla^{2} +m^{2})\delta^{(3)}(\vec x-\vec y)$, after which you use "self-adjointness" of $\nabla^2$ (really, integration by parts), to make the wave operator act on the first $\phi$. You might need to relabel variables afterwards.
A: $$i {{\partial}\over{\partial t}}\pi=[\pi,\int d^3x\tfrac{1}{2}\pi^2+\tfrac{1}{2}\phi()\phi]$$
$$=[\pi,\int d^3x\tfrac{1}{2}\phi()\phi]$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi()\phi]$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi]()\phi+\phi[\pi,()\phi]$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi]()\phi+\phi[\pi,()]\phi+\phi()[\pi,\phi]$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi]()\phi+\{\phi\pi()\phi-\phi()\pi\phi+\phi()\pi\phi-\phi()\phi\pi\}$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi]()\phi+\{\phi\pi()\phi-\phi()\phi\pi\}$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi]()\phi+\{\pi\phi()\phi-\phi\pi()\phi\}$$
$$=\tfrac{1}{2}\int d^3x[\pi,\phi]()\phi+[\pi,\phi]()\phi$$
$$=\int d^3x[\pi,\phi]()\phi$$
$$=\int d^3x-[\phi,\pi]()\phi$$
$$=\int d^3x-i\delta()\phi$$
$$=-i()\phi$$
I'm not sure about the middle part, but I've used a few properties:
(2.44), (2.20), (2.30), and of course the commutator identity. But I don't think that is the correct way of proofing this. (I'm struggling as well)
