The derivative of sin^-1(3x - 4x³) with respect to sin^-1 x is,

3 , for $|x|<\frac{1}{2}$ and -3 for $\frac{1}{2}<|x|<1$

We have,

$y=\sin ^{-1}\left(3 x-4 x^3\right)= \begin{cases}3 \sin ^{-1} x, & \text { if }|x| \leq \frac{1}{2} \\ \pi-3 \sin ^{-1} x, & \text { if } \frac{1}{2}<x \leq 1 \\ -\pi-3 \sin ^{-1} x, & \text { if }-1 \leq x<-\frac{1}{2}\end{cases}$

Let $z=\sin ^{-1} x$. Then,

$y= \begin{cases}3 z & , \text { if }|x| \leq \frac{1}{2} \\ \pi-3 z, & \text { if } \frac{1}{2}<x \leq 1 \\ -\pi-3 z, & \text { if }-1 \leq x<-\frac{1}{2}\end{cases}$

$\Rightarrow \frac{d y}{d z}= \begin{cases}3, & \text { if }|x|<\frac{1}{2} \\ -3, & \text { if } \frac{1}{2}<|x|<1\end{cases}$