Unitary transformation of Dirac equation

Dirac equation is given by $$(igamma^mupartial_mu-m)psi=0.$$ The matrices $gamma^mu$ satisfy the relation $${gamma^mu,gamma^nu}=gamma^mugamma^n+gamma^nugamma^mu=2g^{munu},$$ where
$g^{munu}$ is the Minkowski metric.

I read from Ryder’s QFT textbook (pg 43) that for a $4times 4$ unitary matrix $S$,

1. $gamma’^mu=Sgamma^mu S^{-1}$ also satisfy the the relation ${gamma’^mu,gamma’^nu}=2g^{munu}$.
2. $psi’=Spsi$ also satisfies the Dirac equation if the matrices are transformed using $gamma’^mu=Sgamma^mu S^{-1}$.

How can it be shown that the above two statements are true?

Here are my attempts to prove these two statements:

1. ${gamma’^mu,gamma’^nu}=gamma’^mugamma’^nu+gamma’^nugamma’^mu
   = Sgamma^mu S^{-1}Sgamma^nu S^{-1} + Sgamma^nu S^{-1}Sgamma^mu S^{-1} =Sgamma^mu gamma^nu S^{-1} +
   Sgamma^nu gamma^mu
   S^{-1}=S{gamma^mu,gamma^nu}S^{-1}=2Sg^{munu}S^{-1}$
2. I don’t know how to show this. It seems like the author is doing
   something like a coordinate transformation using the unitary matrix
   $S$. How does $partial_mu$ transform in this case?