In both parts (a) and (b), `1` on the vertical scale is at the top of the figure, where both graphs peak. The sine graph crosses the horizontal axes at multiples of `pi` — thus, from left to right, at `-2pi`, `-pi`, `pi`, and `2pi`. For the approximate derivative graph, the crossings are at odd multiples of `pitext[/]2`, namely, `-3pitext[/]2`, `-pitext[/]2`, `pitext[/]2`, and `3pitext[/]2`.
If `y=sin t`, then the difference quotient `Delta ytext[/]Delta t` that approximates `dytext[/]dt` is
`(sintext[(]t+Delta t text[)]-sintext[(]t text[)])/(Delta t)`.
The formula in part (c) is this difference quotient with `Delta t=0.001`, a small enough number to give a very good approximation to the derivative function. If you guessed that the derivative looks a lot like `cos t`, that's a good guess.