Estimate the tangent by taking the secants you can compute; you the tangent will lie somewhere between the two.
That is: you cannot compute the slope exactly, but you can estimate it from the information you have.
You have immediately preceding and immediately succeeding points, $t_{i-1}$ and $t_{i+1}$, as well as value $f(t_{i-1})$, $f(t_i)$, and $f(t_{i+1})$. The secant between $(t_{i-1},f(t_{i-1}))$, and $(t_i,f(t_i))$ has a certain slope $L_1$; the secant between $(t_i,f(t_i))$ and $(t_{i+1},f(t_{i+1}))$ has a certain slope $L_2$. Unless the graph is very nasty, the tangent at $(t_i,f(t_i))$ should have a slope that is somewhere between $L_1$ and $L_2$.