Covariance function of a differencing in time series $z_t = x_t-x_{t-1}$

Question

I am having trouble finding the covariance function for a differencing of a stationary time series $x_t$.

Letting $z_t = x_t-x_{t-1}$

I see that $E[z_t]=0$ so

$$\gamma_z(h)=E[z_{t+h}z_t]$$ is what I want to simplify.

I understand that

$$\begin{align} \gamma_x(h) & = E[(x_{t+h}-\mu_x)(x_t-\mu_x)]\\ & = E[x_{t+h}x_t]-\mu_x^2\\ \end{align}$$

or in other words

$$E[x_{t+h}x_t]=\gamma_x(h)+\mu_x^2$$

so I am thinking that this should be incorporated in the expression, but all I get is

$$\gamma_z(h) = \gamma_x(h)+\mu_x^2-E[x_{t+h}x_{t-1}]-E[x_{t+h-1}x_t]+E[x_{t-1+h}x_{t-1}]$$

I am just starting to learn time series so it would be very helpful if there is some guidance.

Thank you.