Differentiate and simplify: y = arcsin(x^4), also written y = sin^-1(x^4).

Step-by-step solution

1. 1

   Recognise a composition and pick the right pair of rules. The outer function is the inverse sine and the inner function is $u=x^{4}$, so this needs the arcsine derivative together with the chain rule. Note that $\sin^{-1}$ here means arcsine, not $1/\sin$ — a genuine source of confusion in this notation.

2. 2

   State the arcsine derivative in chain-rule form. If $y=\sin^{-1}(u)$ with $u$ a function of $x$, then

   $\frac{dy}{dx}=\frac{u'}{\sqrt{1-u^{2}}}$

   Here $u=x^{4}$.

3. 3

   Differentiate the inner function. By the power rule,

   $u'=\frac{d}{dx}\left(x^{4}\right)=4x^{3}$

4. 4

   Substitute and simplify the radicand. Putting $u=x^{4}$ and $u'=4x^{3}$ into the formula,

   $\frac{dy}{dx}=\frac{4x^{3}}{\sqrt{1-\left(x^{4}\right)^{2}}}=\frac{4x^{3}}{\sqrt{1-x^{8}}}$

   The key simplification is $\left(x^{4}\right)^{2}=x^{4\cdot 2}=x^{8}$, not $x^{16}$ or $x^{6}$.

5. 5

   Note the domain and verify numerically. The expression requires $1-x^{8}>0$, i.e. $-1<x<1$, which matches the domain of $\sin^{-1}(x^{4})$ minus its endpoints where the derivative blows up. A central difference of $\sin^{-1}(x^{4})$ with $h=10^{-6}$ gives $0.10800354$ at $x=0.3$, $1.41334288$ at $x=0.7$ and $-0.50097943$ at $x=-0.5$, matching $4x^{3}/\sqrt{1-x^{8}}$ to eight decimals at all three points.