First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$

where $$\mu = 0.85 \\ \phi=0.59$$

and $$W_t = X_t - X_{t-1}$$

Compute $$Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(W_t,W_{t-4})=-0.04$$

Hint: use bilinearity $$\text{Cov}\left(W_{t}, W_{t-1} \right) = \text{Cov}\left(X_t - X_{t-1}, X_{t-1} - X_{t-2} \right) = 2\gamma_X(1) - \gamma_X(2) - \gamma_X(0)$$ where $$\gamma_X(h) = \text{Cov}(X_{t+h},X_t)$$ is the autocovariance function of your original process.