sinn (ax2 + bx + c)

Let

$y=\sin ^{n}\left(a x^{2}+b x+c\right)$

Differentiating both sides w.r.t. $x$

$\frac{d y}{d x}=\frac{d}{d x} \sin ^{n}\left(a x^{2}+b x+c\right)$

$=n \cdot \sin ^{n-1}\left(a x^{2}+b x+c\right) \cdot \frac{a}{d x} \sin \left(a x^{2}+b x+c\right)$

$=n \cdot \sin ^{n-1}\left(a x^{2}+b x+c\right) \cdot \cos \left(a x^{2}+b x+c\right) \cdot \frac{d}{d x}\left(a x^{2}+b x+c\right)$

$=n \cdot \sin ^{n-1}\left(a x^{2}+b x+c\right) \cdot \cos \left(a x^{2}+b x+c\right) \cdot(2 a x+b)$

Thus, $\frac{d y}{d x}=n(2 a x+b) \cdot \sin ^{n-1}\left(a x^{2}+b x+c\right) \cdot \cos \left(a x^{2}+b x+c\right)$