$\sin {x^2} + {\sin ^2}x + {\sin ^2}\left( {{x^2}} \right)$

Let $y = \sin {x^2} + {\sin ^2}x + {\sin ^2}\left( {{x^2}} \right)$

therefore,$\frac{{dy}}{{dx}} = \frac{d}{{dx}}\sin \left( {{x^2}} \right) + \frac{d}{{dx}}{(\sin x)^2} + \frac{d}{{dx}}{\left( {\sin {x^2}} \right)^2}$

$= \cos \left( {{x^2}} \right)\frac{d}{{dx}}\left( {{x^2}} \right) + 2\sin x \cdot \frac{d}{{dx}}\sin x + 2\sin {x^2} \cdot \frac{d}{{dx}}\sin {x^2}$

$= \cos {x^2}2x + 2 \cdot \sin x \cdot \cos x + 2\sin {x^2}\cos {x^2} \cdot \frac{d}{{dx}}{x^2}$

$= 2x\cos {(x)^2} + 2 \cdot \sin x \cdot \cos x + 2\sin {x^2} \cdot \cos {x^2} \cdot 2x$

$= 2x\cos {(x)^2} + \sin 2x + \sin 2{(x)^2} \cdot 2x$

$= 2x\cos \left( {{x^2}} \right) + 2x \cdot \sin 2\left( {{x^2}} \right) + \sin 2x$

$\Rightarrow \frac{dy}{dx} = 2x\cos \left( {{x^2}} \right) + 2x \cdot \sin 2\left( {{x^2}} \right) + \sin 2x$