Help deriving Maxwell's equations from the Lagrangian

Question

Starting with the Lagrangian density $$\mathcal{L} = -\frac{1}{2}(\partial_\mu \mathcal{A}_\nu)(\partial^\mu \mathcal{A}^\nu)+\frac{1}{2}(\partial_\mu \mathcal{A}^\mu)^2,$$ I don't understand how to take the partial derivative of the Lagrangian with respect to $\partial(\partial_\mu \mathcal{A}_\nu)$. (Note the answer is: $-\partial^\mu \mathcal{A}^\nu+(\partial_\rho \mathcal{A}^\rho)\eta^{\mu \nu}$)

Answer 1

Writing $\mathcal{L} = - \frac{1}{2} \partial_\rho A_\sigma \partial^\rho A^\sigma + \frac{1}{2}(\partial_\rho A^\rho)^2$ and using the product rule of differentiation $$ \frac{\partial \mathcal{L}}{\partial (\partial_\mu A_\nu)} = - \frac{1}{2}\frac{\partial (\partial_\rho A_\sigma)}{\partial (\partial_\mu A_\nu)} \partial^\rho A^\sigma - \frac{1}{2} \partial_\rho A_\sigma \frac{\partial (\partial^\rho A^\sigma)}{\partial (\partial_\mu A_\nu)} + \frac{1}{2} 2 (\partial_\rho A^\rho) \frac{\partial (\partial_\sigma A^\sigma)}{\partial (\partial_\mu A_\nu)} $$ Now $$ \frac{\partial (\partial_\rho A_\sigma)}{\partial (\partial_\mu A_\nu)} = \delta^\mu_\rho \delta^\nu_\sigma \quad \quad \frac{\partial (\partial^\rho A^\sigma)}{\partial (\partial_\mu A_\nu)} = \eta^{\mu\rho} \eta^{\nu \sigma} \quad \quad \frac{\partial (\partial_\sigma A^\sigma)}{\partial (\partial_\mu A_\nu)} = \delta^\mu_\sigma \eta^{\sigma \nu} = \eta^{\mu \nu} $$ The second and third equalities follow from writing for example $\partial^\rho A^\sigma = \eta^{\rho \alpha} \eta^{\sigma \beta} \partial_\alpha A_\beta $ and using the first equality. Using them, you obtain $$ \frac{\partial \mathcal{L}}{\partial (\partial_\mu A_\nu)} = - \partial^\mu A^\nu + \eta^{\mu \nu} \partial_\rho A^\rho $$ which gives you Maxwell equation.