Here's another way to think about this .

We want to know the speed at exactly t equals 10.

Speed is delta s, over delta t.

We would like delta t to be 0,

so, the only t we're dealing with is t equals 10,

but we can't have delta t be 0,

because division by 0

leads to a quantity that diverges, is not defined,

just doesn't make sense,

and so we just sneak up on dividing by 0,

we let our delta t get

smaller, and smaller, and smaller, and smaller,

so we get closer, and closer, and closer

to dividing by 0,

this ratio will stay constant in most circumstances

and we say that that's the instantaneous speed.

As you probably guessed,

this instantaneous speed

or in general, this instantaneous rate of change,

is known as a derivative.

I'll define the derivative more carefully in a moment,

but first I want to give a graphical picture

to accompany the stories

and numbers I talked about here.