sin x2 + sin2 x + sin2 (x2)
Question:

sin x2 + sin2 x + sin2 (x2)

Solution:

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

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

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

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

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

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

Thus, $\frac{d y}{d x}=2 x \cdot \cos x^{2}+\sin 2 x+2 x \sin 2 x^{2}$

 

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