Video Transcript
Let 𝑔 of 𝑥 be equal to arcsin of
𝑥 to the power of four cubed. Find 𝑔 prime of 𝑥.
𝑔 of 𝑥 is expressed as a function
of a function of a function. It’s a composite function. We’re therefore going to need to
use the chain rule to find the derivative 𝑔 prime of 𝑥. The chain rule says that if 𝑦 is a
function of 𝑢 and 𝑢 is a function of 𝑥, then d𝑦 by d𝑥 is equal to d𝑦 by d𝑢
times d𝑢 by d𝑥.
A special case of the chain rule is
the general power rule. And this says that if 𝑢 is a
function of 𝑥, then the derivative of 𝑢 to the power of 𝑛 can be written as 𝑛
times 𝑢 to the power of 𝑛 minus one multiplied by the derivative of 𝑢 with
respect to 𝑥.
We’re actually going to apply both
of these rules during this question. We can use the general power rule
to begin finding 𝑔 prime of 𝑥. Since our function in 𝑥 is arcsin
of 𝑥 to the power of four and then that’s being cubed, we can say that 𝑔 prime of
𝑥 must be equal to three times that function arcsin of 𝑥 to the power of four, and
then that’s squared. And we multiply that by the
derivative of arcsin of 𝑥 to the power of four with respect to 𝑥.
So we’re going to need to use the
chain rule to actually evaluate the derivative of arcsin of 𝑥 to the power of four
with respect to 𝑥. We’ll say that 𝑦 is equal to
arcsin of 𝑢 and 𝑢 is equal to 𝑥 to the power of four.
To use the chain rule, we’re going
to need to find the derivative of each of these. The derivative of 𝑢 with respect
to 𝑥 is four 𝑥 cubed. We’ll also use the fact that the
derivative of arcsin of 𝑥 with respect to 𝑥 is one over the square root of one
minus 𝑥 squared. This means that d𝑦 by d𝑢 is one
over the square root of one minus 𝑢 squared.
We substitute this back into the
formula for the chain rule. And we see that the derivative of
arcsin of 𝑥 to the power of four with respect to 𝑥 is one over the square root of
one minus 𝑢 squared times four 𝑥 cubed. Remember that we’re trying to
differentiate this with respect to 𝑥. So we’re going to use the fact that
we let 𝑢 be equal to 𝑥 to the power of four. And when we substitute this back
into the expression for the derivative, we get four 𝑥 cubed over the square root of
one minus 𝑥 to the power of four all squared.
Now, in fact, 𝑥 to the power of
four squared is 𝑥 to the power of eight. And we can replace this in our
original equation for 𝑔 prime of 𝑥. And we get three arcsin of 𝑥 to
the power of four squared times four 𝑥 cubed over the square root of one minus 𝑥
to the power of eight.
Simplifying just a little, and we
find that 𝑔 prime of 𝑥, the derivative of our function 𝑔 with respect to 𝑥, is
12 arcsin of 𝑥 to the power of four squared over the square root of one minus 𝑥 to
the power of eight.
