So let me now define the derivative

a little bit more formally.

Returning to the example we started with,

average speed, from time 1 to time 2

is the distance travelled, during that time interval,

divided by the time interval itself,

and then we can say, well, let t2,

the second time, get closer and closer to t1,

that's equivalent to saying

if I think about average speed as this:

change in position, over a change in time,

let this time interval, delta t, get closer and closer to 0,

so, if I'm thinking in terms of this notation:

delta s over delta t,

I let delta t get closer and closer and closer to 0

I can't just plug in 0 because that'll be dividing by 0,

but I let it get closer and closer and closer to 0

and I monitor what this number becomes,

what this ratio becomes,

and the result is the instantaneous speed,

and the notation for that often is: ds over dt,

so instead of these triangles - delta's, which means big change,

we use these little d's - which mean: instantaneous change.

So, finally, and a little bit more generally,

let's think about the derivative of x

so here x could be any function that changes at all.

this could be the position, it could be temperature, it could be population,

it's just some generic function that changes,

it doesn't have to be position,

and we could ask: How does it change instantaneously?

So, the derivative of x, dx/dt,

is the instantaneous rate of change at x

that's the key thing you need to know from this subunit:

the derivative is the instantaneous rate of change.

Let me write just a few other ways to picture this -

it's also the slope of x of t. x of t could be a curve -

but that's okay - we saw on the previous video

how to think about slopes have curves,

and, for those of you who have had calculus

here's the formal definition:

it's the limit as h goes to 0

x of t plus h, minus x of t over h,

which I just like to think of as:

change in x, over change in t,

in the limit that these changes become very, very small.

So, again, the derivative is:

the instantaneous rate of change of x

It's telling you how fast x is changing

and, you might get different values, depending on the time

so this allows for a function x that's growing

at different rates, at different times.

Lastly, just a little bit notation:

so this idea of a derivative

is quite often

to note it this way: which should be read x-prime,

and in physics and engineering,

one often uses this notation: x-dot,

which the stands for time derivative.

In this course are almost always use this first derivative.

I might occasionally, maybe by mistake, use this one,

I won't use this one at all, but you're likely to see this

in other texts or papers particularly if it's from

the physics or engineering community.