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Derivatives on tensor components

Problem Statement
I want to prove that [tex] -\frac{\partial}{\partial p_\mu}\frac{1}{\not{p}}=\frac{1}{\not{p}}\gamma^\mu \frac{1}{\not{p}}[/tex]
Relevant Equations
[tex] \not{p}=\gamma^\mu p_\mu[/tex]
This was my attempt at a solution and was wondering where did I go wrong: [tex] -\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}[/tex]
Any hint would be great! Thank you!


Science Advisor
Insights Author
Apply the Leibniz rule to [tex]\left( \frac{\partial}{\partial p_{\mu}} \frac{1}{\not\! p} \right) \not \! p \ .[/tex]

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