Where Did I Go Wrong in Deriving Tensor Component Derivatives?

[RicardoMP]

Homework Statement

I want to prove that $$-\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=\frac{1}{\not{p}}\gamma^\mu \frac{1}{\not{p}}$$

Relevant Equations

$$\not{p}=\gamma^\mu p_\mu$$

This was my attempt at a solution and was wondering where did I go wrong: $$-\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=-\frac{\partial}{\partial p_\mu}[\gamma^\nu p_\nu]^{-1}=\gamma^\nu\frac{\partial p_\nu}{\partial p_\mu}[\gamma^\sigma p_\sigma]^{-2}=\gamma^\nu\delta^\nu_\mu\frac{1}{\not{p}^2}=\gamma^\mu\frac{1}{\not{p}^2}$$
Any hint would be great! Thank you!

 

[samalkhaiat]

Apply the Leibniz rule to $$\left( \frac{\partial}{\partial p_{\mu}} \frac{1}{\not\! p} \right) \not \! p \ .$$