Start from the fundamental relation:

(1)
\begin{equation} S(E,V,N) \end{equation}

Define the Legendre transform that replaces $E$ with $1/T$ and $N$ with $\mu /T$:

(2)
\begin{align} \psi = S-\frac{E}{T}+\frac{\mu}{T}N \end{align}

then

(3)
\begin{align} d\psi = dS - Ed(\frac{1}{T})-\frac{1}{T}dE+\frac{\mu}{T}dN+Nd\left(\frac{\mu}{T}\right) \end{align}

Now use $dS = (1/T)dE+(P/T)dV-(\mu/T) dN$ to get

(4)
\begin{align} d\psi = \frac{P}{T}dV-Ed(\frac{1}{T})-Nd\left(\frac{\mu}{T}\right) \end{align}

We can now write the following Maxwell relation for this fundamental relation:

(5)
\begin{align} \left(\frac{\partial(P/T)}{\partial(1/T)}\right)_{V,\mu/T} = -\left(\frac{\partial E}{\partial V}\right)_{\mu/T,1/T} \end{align}

Now carry out the differentiation on the left hand side:

(6)
\begin{align} P+\frac{1}{T}\left(\frac{\partial P}{\partial(1/T)}\right)_{V,\mu/T} = -\left(\frac{\partial E}{\partial V}\right)_{\mu/T,1/T} \end{align}

which the desired result.