### Video Transcript

Let π of π₯ be equal to one-half
π₯ cubed plus one-half π₯ squared plus five π₯ minus four and let π be the inverse
of π. Given that π of two is equal to
12, what is π prime of 12?

In order to help us find π prime
of 12, we can use the formula for derivatives of inverse functions. This tells us that if π is the
inverse function of π, then π prime of π¦ is equal to one over π prime of π of
π¦. Letβs start by finding π prime of
π₯, the derivative of π with respect to π₯. We can see that π is a
polynomial. Therefore, in order to find its
derivative, we can differentiate it term by term using the power rule for
differentiation. We simply multiply by the power and
decrease the power by one. This gives us that π prime of π₯
is equal to three over two π₯ squared plus π₯ plus five.

Next, weβll observe the fact that
weβre trying to find π prime of 12. And so, we can substitute π¦ equals
12 into our formula for π prime of π¦. This gives us that π prime of 12
is equal to one over π prime of π of 12. Now, we do not know what π of 12
is. However, we have been given in the
question that π of two is equal to 12. And since π is the inverse
function of π, we can apply π to both sides here. And weβll obtain that π of 12 is
equal to two. This is because of the way inverse
functions work. If we take π of π of two, then
weβll simply get two.

We can now substitute this value of
π of 12 back into our equation for π prime of 12. And we obtain that itβs equal to
one over π prime of two. Now, weβve already found π prime
of π₯. So we can simply substitute π₯
equals two in order to find π prime of two. And we obtain that π prime of two
is equal to 13. And substituting the value of π
prime of two back into π prime of 12, we obtain that π prime of 12 is equal to one
over 13.