# Video: AP Calculus AB Exam 1 • Section I • Part A • Question 22

Let 𝑔(𝑥) = (arcsin (𝑥⁴))³. Find 𝑔′(𝑥).

02:50

### 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.