Let f of x equal two x squared.
We want to calculate f prime of one,
or find the derivative
function value at x equals one
using the limit definition
of the derivative,
which is given here.
Where f prime of one will give us
the slope of the tangent
line at x equals one,
which we'll verify
graphically when we finish.
So to find our derivative
function value at x equals one,
we'll substitute one for
x here, here and here.
So, f prime of one is equal to the limit
as h approaches zero of
our difference quotient, which would be
f of the quantity one plus h,
minus f of one
all divided by h.
This will give us the limit
as h approaches zero of,
notice here the input into our function
is the quantity one plus h.
So, we'd have two times the
quantity one plus h squared,
minus f of one, which would
be two times one squared,
divided by h.
Next we need to square
the quantity one plus h.
There's no shortcuts here.
We have to write out two
factors of one plus h,
and we'll have four products.
One, two, three and four.
So, we have one times one is one.
Then one times h is h.
Plus h times one, that's also h.
So, plus two h.
Then plus h times h or plus h squared.
So, we have two times the quantity
one plus two h, plus h squared,
minus two times one squared is just two,
all over h.
Now, we'll distribute the two.
So we have the limit as
h approaches zero of,
distributing two, we have two plus four h,
plus two h squared.
We still have minus two all over h.
Notice numerator does simplify.
Two minus two simplifies to zero,
which now our numerator is
just four h plus two h squared,
and our denominator is h.
Let's continue on the next slide.
There's a couple of ways to
simplify this fraction here.
Because we are dividing by a monomial,
we can write this as the
limit as h approaches zero of
four h divided by h
plus two h squared divided by h,
and then simplify each
fraction individually,
which would give us the limit
as h approaches zero of.
Notice here h over h simplifies to one,
so we have just four,
plus here we have a common factor of h.
This simplifies to one.
Here we have just one factor of h.
So, we have the limit
as h approaches zero of
four plus two h.
So, this would be one way
to simplify this fraction.
The other way would be
to factor the numerator.
Notice how there's a common factor of h.
So, we'd have h times the
quantity four plus two h
and divide it by h
and here we can simplify these h's.
Again, leaving us with the
quantity four plus two h.
Then finally as h approaches zero,
two h approaches zero
and therefore our limit is
going to equal positive four.
So, f prime of one is
equal to positive four
meaning our derivative
function value is four
when x equals one,
which again does give us the
slope of the tangent line
to our function at x equals one.
And let's verify this graphically.
In blue, we have the graph of
f of x equals two x squared.
When x equals one, the
point of the function
is the point one comma
two, this point here.
So, this red line is our
tangent line at that point
and if we select two points
on this tangent line,
we can verify the slope
of this tangent line.
So, we'll use the point of tangency here
and let's use this point here.
Notice how this point would
have coordinates two comma six.
If you want to move from
the point on the left
to the point on the right,
notice how we'd have to go up four units
and then right one unit.
Verifying the slope of the tangent line
at that point is positive four.
I hope you found this helpful.
