- [Voiceover] What you see here in blue
this is the graph of
Y is equal to F of X.
Where F of X is equal to
X to the third
minus six X squared
plus X minus five.
What I want to do in this video
is think about what is the equation
of the tangent line
when X is equal to one?
So we can visualize that.
So, this is X equaling one right over here.
This is the value of the function.
When X is equal to one.
Right over there.
And then the tangent line
looks something like
will look something like.
I know I can do a better job than that.
It's going to look something like that.
And what we want to do is find the equation
the equation of that line.
And if you are inspired I encourage you to be,
pause the video and try to work it out.
Well the way that we can do this is
if we find the derivative at X equals one
the derivative is the slope of the tangent line.
And so we'll know the slope of the tangent line.
And we know that it contains that point
and then we can use that to find the equation
of the tangent line.
So let's actually just, let's just.
So we want the equation of the tangent line
when X is equal to one.
So let's just first of all evaluate F of one.
So F of one
is equal to one to the third power
which is one.
Minus six times ones squared,
so it's just minus six.
And then
plus one
minus five.
So, this is equal to what?
Two minus 11?
Which is equal to
negative nine.
And that looks about right.
That looks like about negative nine right over there.
The scales are different on the Y and the X axis.
So that is F of one.
It is negative nine.
Did I do that right?
This is negative five.
Yep, negative nine.
And now let's evaluate what the derivative is
at one.
So,
what is F prime of X?
F prime of X.
Well here it's just a polynomial.
You take the derivative of X to the third
while we apply the power rule.
We bring the three out front.
So you get three X
to the.
And then we go one less than three
to get the second power.
Then you have
minus six X squared.
So you bring the two times the six
to get 12.
So minus 12 X
to the
well two minus one is one power
so that's the same thing as 12 X.
And then plus the derivative of X
is just one.
That's just going to be one.
And if you view this as X to the first power
we're just bringing the one out front
and decrementing the one.
So it's one times X to the zero power
which is just one.
And then the derivative of a constant here
is just going to be zero.
So this is our derivative of F
and if we want to evaluate it at one
F prime of one
is going to be three times one squared
which is just three
minus 12 times one
which is just minus 12.
And then we have plus one.
So this is
three minus 12 is negative nine
plus one is equal to negative eight.
So we know the slope right over here
is the slope of negative eight.
We know a point on that line
it contains the point one negative nine
so we can use that information to find the
equation of the line.
The line, just to remind ourselves,
has the four.
Y is equal to M X plus B.
Where M is the slope.
So we know that Y is going to be
equal to negative eight X
plus B.
And now we can substitute
the X and Y value that we know
sits on that line to solve for B.
So we know that Y is equal to negative nine.
Let me just write this here.
Y is equal to negative nine
when X is equal to
when X is equal to one.
And so we get
we get
negative nine
is equal to negative eight times one.
So negative eight
plus B.
Well, let's see.
We could add
we could eight to both sides
and we get negative one
is equal to B.
So we're done.
The equation of the line
the equation of this line
that we have in magenta right over there
that is
Y is equal to the slope
is negative eight X.
And then the Y-intercept
minus one.