is a few examples that test our intuition

of the derivative as a rate of change

or the steepness of a curve

or the slope of a curve

or the slope of a tangent line of a curve

depending on how you actually want to think about it.

So here it says F prime of five

so this notation, prime

this is another way of saying well what's the derivative

let's estimate the derivative of our function at five.

And when we say F prime of five this is the slope

slope of tangent line

tangent line at five

or you could view it as the

you could view it as the rate of change

of Y with respect to X

which is really how we define slope

respect to X of our function F.

So let's think about that a little bit.

We see they put the point

the point five comma F of five right over here

and so if we want to estimate the slope of the tangent line

if we want to estimate the steepness of this curve

we could try to draw a line

that is tangent right at that point.

So let me see if I can do that.

So if I were to draw a line starting there

if I just wanted to make a tangent

it looks like it would do something like that.

Right at that point that looks to be

about how steep that curve is

now what makes this an interesting thing in non-linear

is that it's constantly changing

the steepness it's very low here and it gets steeper

and steeper and steeper as we move to the right

for larger and larger X values.

But if we look at the point in question

when X is equal to five remember F prime of five

would be if you were estimating it

this would be the slope of this line here.

And the slope of this line it looks like

for every time we move one in the X direction

we're moving two in the Y direction.

Delta Y is equal to two when delta X is equal to one.

So our change in Y

with respect to X

at least for this tangent line here

which would represent our change in Y

with respect to X right at that point

is going to be equal to two over one, or two.

And it's almost estimated, but all of these are way off.

Having a negative two derivative

would mean that as we increase our X our Y is decreasing.

So if our curve looks something like this

we would have a slope of negative two.

If having slopes in this

a positive of point one

that would be very flat something down here

we might have a slope closer to point one.

Negative point one that might be closer on this side

now we're sloping but very close to flat.

A slope of zero, that would be right over here at the bottom

where right at that moment as we change X

Y is not increasing or decreasing

the slope of the tangent line right at that bottom point

would have a slope of zero.

So I feel really good about that response.

Let's do one more of these.

So alright, so they're telling us to compare

the derivative of G at four to the derivative of G at six

and which of these is greater

and like always, pause the video

and see if you can figure this out.

Well this is just an exercise

let's see if we were to

if we were to make a line that indicates the slope there

you can do this as a tangent line

let me try to do that.

So now that wouldn't, that doesn't do a good job

so right over here at

that looks like a

I think I can do a better job than that

no that's too shallow to see

not shallow's not the word, that's too flat.

So let me try to really

okay, that looks pretty good.

So that line that I just drew

seems to be indicative of

the rate of change of Y with respect to X

or the slope of that curve

or that line you can view it as a tangent line

so we could think about what its slope is going to be

and then if we go further down over here

this one is, it looks like it is

steeper but in the negative direction

so it looks like it is steeper for sure

but it's in the negative direction.

As we increase, think of it this way

as we increase X one here

it looks like we are decreasing Y by about one.

So it looks like G prime of four

G prime of four, the derivative when X is equal to four

is approximately, I'm estimating it

negative one

while the derivative here when we increase X

if we increase X by

if we increase X by one

it looks like we're decreasing Y by close to three

so G prime of six

looks like it's closer to negative three.

So which one of these is larger?

Well, this one is less negative

so it's going to be greater than the other one

and you could have done this intuitively

if you just look at the curve

this is some type of a sinusoid here

you have right over here the curve is flat

you have right at that moment

you have no change in Y with respect to X

then it starts to decrease

then it decreases at an even faster rate

then it decreases at a faster rate

then it starts, it's still decreasing

but it's decreasing at slower and slower rates

decreasing at slower rates and right at that moment

you have your slope of your tangent line is zero

then it starts to increase, increase, so on and so forth

and it just keeps happening over and over again.

So you can also think about this in a more intuitive way.