Prove function is differentiable

Let $\displaystyle f$ be continuous on the reals and let $\displaystyle F(x) = \int_{x-1}^{x+1}f(t)dt$. Show that $\displaystyle F$ is differentiable on the reals and compute $\displaystyle F'$.

So this should seem straightforward but I'm having trouble actually formalizing it simply because the lower and upper limits of the integral are both variables. I thought of saying that $\displaystyle F(x) = \int_{0}^{x+1}f(t)dt - \int_{0}^{x-1}f(t)dt$ but I'm not sure how to continue from there via utilizing the FTC. Any help would be appreciated.