Hello, kingsolomonsgrave!
$\displaystyle \frac{dy}{dx} \;=\;\frac{6x-\frac{1}{2}x^{-\frac{1}{2}}}{\sin(y)} \quad\Rightarrow\quad \frac{dy}{dx} \;=\; \frac{12x^{\frac{3}{2}}-1}{2\sqrt{x}\sin(y)}\quad\text{How?}$
We have: .$\displaystyle \frac{6x-\frac{1}{2}x^{-\frac{1}{2}}}{\sin(y)} $
Multiply by$\displaystyle \frac{2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}\!:\;\; \frac{2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}\cdot \frac{6x-\frac{1}{2}x^{-\frac{1}{2}}}{\sin(y)} \;=\;\frac{12x^{\frac{3}{2}} - 1}{2x^{\frac{1}{2}}\sin(y)} $