We're asked to use the limit definition
of the derivative to find the derivative function value
of f of x equals three divided by x and x equals four,
which means we want to determine f prime of four.
And there are two ways to do this.
One way would be to find the derivative function
by using the limit definition of the derivative,
shown here in the lower left,
and then substitute four for x,
or we can find the derivative function value
in one step by substituting four for x
into the limit definition of the derivative.
So if we substitute four for x here, here, and here,
we have f prime of four is equal to the limit
as h approaches zero of the quantity f
of four plus h minus f of four,
all divided by h.
And let's go ahead and take this approach.
So once again, f prime of four is equal to the limit
as h approaches zero of f of the quantity four plus h
minus f of four, all divided by h.
The next step is to determine f
of the quantity four plus h and f of four.
To find f of the quantity four plus h,
we substitute four plus h into the function,
or substitute four plus h for x,
which gives us three divided by the quantity four plus h,
and then we have minus f of four would be 3/4,
all divided by h.
The next step is to subtract the fractions in the numerator.
In order to subtract or add fractions,
we must have a common denominator,
or the least common denominator.
Four plus h and four only have a common factor of one,
and therefore the LCD, or least common denominator,
is equal to the product of the two denominators,
which gives us four times the quantity four plus h.
So looking at the first fraction
of three divided by the quantity four plus h,
that denominator is missing the factor of four,
and therefore we multiply the numerator
and denominator by four.
The second fraction is missing the factor
of four plus h in the denominator,
and therefore we multiply the numerator
and denominator by four plus h.
Notice now we do have a common denominator of four times
the quantity four plus h.
This gives us the limit as h approaches zero of,
we have an h in the denominator.
And now in the numerator,
we have a common denominator of four times
the quantity four plus h.
And subtracting, we have three times four, which is 12,
and then minus three times the quantity four plus h.
And now simplify this numerator.
We have the limit as h approaches zero.
The denominator is still h.
The denominator of the numerator is still four times
the quantity four plus h
and the numerator is now 12.
Distributing negative three,
we have minus 12 minus three h.
Continuing, we have the limit as h approaches zero.
And now let's focus on the numerator,
where the numerator has a denominator
of four times the quantity four plus h,
and the numerator is now 12 minus 12 minus three h,
which gives us negative three h.
And we still have a denominator here of h.
And remember, a fraction bar means division.
Dividing by h is equivalent
to multiplying by the reciprocal,
which would be the same as multiplying by one over h.
So we'll go ahead and multiply by one over h.
And now before multiplying,
notice how we have a common factor of h between
the numerator and denominator, here and here,
that simplifies to one,
which leaves us with the limit as h approaches zero
of negative three divided by four times
the quantity four plus h.
And now we can determine the limit
as h approaches zero.
We just have negative three divided by four times four.
Four times four is 16.
So now we know f prime of four is equal to negative 3/16,
which means this is the slope of the tangent line
to the graph of f of x at x equals four.
And let's verify this graphically.
In blue, we have the graph of f of x equals three
divided by x.
At x equals four, this is the point on the graph,
which is four comma 3/4,
and the slope of this red tangent line at x equals four
is negative 3/16, verifying we did find
the derivative function value correctly.
I hope you've found this helpful.