### Video Transcript

Given that π¦ is equal to π of π₯
is a function for four known values, where π of two is equal to three, π of six is
equal to 3.75, π of seven is equal to four, and π of 11 is equal to 4.25, estimate
π prime of seven.

In this question, weβve been asked
to estimate the derivative of π at seven and weβve been given π of π₯ values near
seven. Therefore, we can use the numerical
method in order to estimate this derivative. We have that π prime of π is
roughly equal to π of π minus π of π over π minus π plus π of π minus π of
π over π minus π all over two, where π is less than π, which is less than
π. And we want to choose the closest
possible values to π for π and π. In our case, since weβre trying to
find π prime of seven, π is equal to seven. And the closest π₯-values on either
side of seven, for which weβve been given their π values, is six and 11. So we can let six be equal to π
and 11 be equal to π.

Next, we can simply substitute
these values into our formula. We have that π prime of seven is
roughly equal to π of seven minus π of six over seven minus six plus π of 11
minus π of seven over 11 minus seven all over two. Now, we know the values of π of
six, π of seven, and π of 11 since theyβve been given to us in the question. So we can substitute these values
in, which leaves us with this. And next, we can simplify the
fractions in the numerator to give us 0.25 over one plus 0.25 over four all over
two.

Now, we can write 0.25 over one as
0.25. And we can write 0.25 over four as
one-fourth multiplied by 0.25. Next, we can rewrite the 0.25s as
one-fourth. And next, we can multiply through
and then add the two fractions in the numerator and then finally divide five over 16
by two to give us that π prime of seven is roughly equal to five over 32. Our solution can also be written in
decimal form as π prime of seven is approximately equal to 0.15625.