### Video Transcript

Let π of π₯ be equal to arcsin of
π₯ to the power of four cubed. Find π prime of π₯.

π of π₯ is expressed as a function
of a function of a function. Itβs a composite function. Weβre therefore going to need to
use the chain rule to find the derivative π prime of π₯. The chain rule says that if π¦ is a
function of π’ and π’ is a function of π₯, then dπ¦ by dπ₯ is equal to dπ¦ by dπ’
times dπ’ by dπ₯.

A special case of the chain rule is
the general power rule. And this says that if π’ is a
function of π₯, then the derivative of π’ to the power of π can be written as π
times π’ to the power of π minus one multiplied by the derivative of π’ with
respect to π₯.

Weβre actually going to apply both
of these rules during this question. We can use the general power rule
to begin finding π prime of π₯. Since our function in π₯ is arcsin
of π₯ to the power of four and then thatβs being cubed, we can say that π prime of
π₯ must be equal to three times that function arcsin of π₯ to the power of four, and
then thatβs squared. And we multiply that by the
derivative of arcsin of π₯ to the power of four with respect to π₯.

So weβre going to need to use the
chain rule to actually evaluate the derivative of arcsin of π₯ to the power of four
with respect to π₯. Weβll say that π¦ is equal to
arcsin of π’ and π’ is equal to π₯ to the power of four.

To use the chain rule, weβre going
to need to find the derivative of each of these. The derivative of π’ with respect
to π₯ is four π₯ cubed. Weβll also use the fact that the
derivative of arcsin of π₯ with respect to π₯ is one over the square root of one
minus π₯ squared. This means that dπ¦ by dπ’ is one
over the square root of one minus π’ squared.

We substitute this back into the
formula for the chain rule. And we see that the derivative of
arcsin of π₯ to the power of four with respect to π₯ is one over the square root of
one minus π’ squared times four π₯ cubed. Remember that weβre trying to
differentiate this with respect to π₯. So weβre going to use the fact that
we let π’ be equal to π₯ to the power of four. And when we substitute this back
into the expression for the derivative, we get four π₯ cubed over the square root of
one minus π₯ to the power of four all squared.

Now, in fact, π₯ to the power of
four squared is π₯ to the power of eight. And we can replace this in our
original equation for π prime of π₯. And we get three arcsin of π₯ to
the power of four squared times four π₯ cubed over the square root of one minus π₯
to the power of eight.

Simplifying just a little, and we
find that π prime of π₯, the derivative of our function π with respect to π₯, is
12 arcsin of π₯ to the power of four squared over the square root of one minus π₯ to
the power of eight.