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## Derivative definition

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# Limit expression for the derivative of function (graphical)

## Video transcript

With the graph of the
function f as an aid, evaluate the following limits. So the first one is the
limit as x approaches 3 of f of x minus f
of 3 over x minus 3. So let's think about x minus--
x equals 3 is right over here. This right over here is f
of 3, or we could say f of 3 is 1 right over here. That's the point 3 comma f of 3. And they're essentially
trying to find the slope between an
arbitrary x and that point as that x gets
closer and closer to 3. So we can imagine an
x that is above 3, that is, say, right over here. Well, if we're trying to find
the slope between this x comma f of x and 3 comma
f of 3, we see that it gets this
exact same form. Your end point is f of x. So it's f of x minus
f of 3 is your change in the vertical axis. That's this distance
right over here. And we would divide
by your change in the horizontal axis,
which is your change in x. And that's going
to be x minus 3. So that's the exact expression
that we have up here when I picked this as an arbitrary x. And we see that that
slope, just by looking at the line between
those two intervals, seems to be negative 2. And the slope was the same
thing if we go the other side. If x was less than
3, then we also would have a slope
of negative 2. Either way, we have a
slope of negative 2. And that's important
because this limit is just the limit
as x approaches 3. So it can be as x approaches
3 from the positive direction or from the negative direction. But in either case, the slope,
as we get closer and closer to this point right over
here, is negative 2. Now let's think about what
they're asking us here. So we have 8, f of 8. So let's think. We have 8. This is 8 comma f of 8. So that's 8 comma f
of 8 right over there. And they have f of 8 plus h. So our temptation
might be to say, hey, 8 plus h is going to
be someplace out here. It's going to be
something larger than 8. But notice, they
have the limit as h approaches 0 from the
negative direction. So approaching 0 from
the negative direction means you're coming
to 0 from below. You're at negative 1,
negative 0.5, negative 0.1, negative 0.0001. So h is actually going
to be a negative number. So 8 plus h would actually
be-- we could just pick an arbitrary point. It could be something
like this right over here. So this might be the
value of 8 plus h. And this would be the
value of f of 8 plus h. So once again, they're
finding-- or this expression is the slope between
these two points. And then we are taking the
limit as h approaches 0 from the negative direction. So as h gets closer
and closer to 0, this down here moves further
and further to the right. And these points move closer
and closer and closer together. So this is really
just an expression of the slope of
the line, roughly-- and we see that it's constant. So what's the slope of the
line over this interval? Well, you can just eyeball
it and see, well, look. Every time x changes by 1,
our f of x changes by 1. So the slope of the
line there is 1. It would have been a
completely different thing if this said limit
as h approaches 0 from the positive direction. Then we would be looking
at points over here. And we would see that we would
slowly approach, essentially, a vertical slope, kind
of an infinite slope.