Let f of x equal four divided by x,
or four over x.
We want to calculate f prime of two,
in order to determine the derivative function
value at x equals two
using the limit definition of the derivative,
which is given here.
Where f prime of two will give
the slope of the tangent line at x equals two,
which we'll verify graphically once we finish.
So, to find f prime of two, using the limit definition,
we'll substitute two for x here, here and here.
Which means f prime of two is equal to the limit
as h approaches zero of
our difference quotient which would be
the f of the quantity two plus h,
minus f of two
divided by h.
And now the term in our function values in the numerator.
So, we have the limit
as h approaches zero of
f of the quantity two plus h would be four,
divided by the quantity two plus h
then we have minus f of two,
f of two would be four divided by two, or just two.
And here we just have h,
and now we'll find this difference in the numerator,
so we'll write two over one
and obtain a common denominator.
So, we have the limit
as h approaches zero of ...
Again, we'll leave this first fraction as
four over the quantity two plus h,
and we'll write minus two as two over one,
this is all over h.
Notice here the common denominator
would be the quantity 2 plus h.
So, multiply two over one by two plus h,
over two plus h.
So, this gives us the limit,
as h approaches zero of ...
Notice here have a common denominator of
the quantity two plus h,
and the numerator here is going to be four,
and then because of the subtraction
we'll distribute negative two.
So, negative two times two to
the negative four minus four
negative two times h is negative two h
or minus two h.
Our denominator here is still h,
but notice how the fraction in the
numerator does simplify
since four minus four is equal to zero.
So notice how the fraction in the
numerator is now just negative two h
over the quantity two plus h.
Let's continue on the next slide.
Now, we want to simplify this complex fraction.
Remember a fraction bar means division,
so we can write this as the limit
as h approaches zero of
the fraction in the numerator would be
negative two h divided by the quantity two plus h,
and then instead of dividing by h
we can multiply by the reciprocal.
The reciprocal of h would be one over h.
So, dividing by h is equivalent to
multiplying by one over h,
and now before we find this product, we can simplify,
h divided by h simplifies to one
which gives us the limit
as h approaches zero of ...
Here we just have negative two
divided by the quantity two plus h,
and now we can just determine our limit.
Notice as h approaches zero
this approaches negative two over positive two,
which equals negative one.
So, f prime of two is equal to negative one,
which again does give us a
slope of the tangent line to our function
at x equals two.
Let's verify this graphically.
In blue we have the graph of our function
f of x equals four divided by x
when x equals two, the point of the
function is this point here, two comma two.
So notice how this red line is the
tangent line at that point.
So, if we select the second point on our tangent line,
we can determine the slope of our tangent line.
So, for example if we select this point here,
notice how to move from the point
on the left to the point on the right,
we'd have to go down one unit
and right one unit.
Verifying this about our tangent line
as negative one divided by one,
or just negative one.
I hope you found this helpful.