Checking a commutation relation for angular momentum and lin. momentum

[tomwilliam2]

Homework Statement

Prove the commutation relation $\left [L_x, p_y \right] = i\hbar \epsilon_{xyz} p_z$

Homework Equations

$L_x = yp_z - zp_y$
$p_z = i\hbar \frac{\partial}{\partial z}$

The Attempt at a Solution

$\left [L_x, p_y \right] = (yp_z - zp_y)p_y - p_y(yp_z - zp_y)$
$\left [L_x, p_y \right] = \left (y\cdot i\hbar \frac{\partial}{\partial z}\cdot i\hbar \frac{\partial}{\partial y} - z\cdot i\hbar \frac{\partial}{\partial y}\cdot i\hbar \frac{\partial}{\partial y}\right ) -\left ( i\hbar \frac{\partial y}{\partial y}i\hbar \frac{\partial}{\partial z} - i\hbar \frac{\partial z}{\partial y} i\hbar \frac{\partial}{\partial y} \right )$
$\left [L_x, p_y \right] = -y\hbar^2 \frac{\partial^2}{\partial z \partial y} + z\hbar^2 \frac{\partial^2}{\partial y} + \hbar^2\frac{\partial^2}{\partial z^2}$

I don't see how I can turn this into the final answer, given that I have derivatives of y still left...so where did I go wrong?
Thanks in advance

 

[hexidecimel10]

You need to remember to use the product rule when taking derivatives.

 

[tomwilliam2]

Thanks! That old mistake...