Question
Differentiate the following w.r.t. x:
$\sin\text{x}^2+\sin^2\text{x}+\sin^2(\text{x}^2)$
$\sin\text{x}^2+\sin^2\text{x}+\sin^2(\text{x}^2)$
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Answer
Let $\text{y}=\sin\text{x}^2+\sin^2\text{x}+\sin^2(\text{x}^2)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\sin\big(\text{x}^2\big)+\frac{\text{d}}{\text{dx}}\big(\sin\text{x}\big)^2+\frac{\text{d}}{\text{dx}}(\sin\text{x}^2)^2$
$=\cos\big(\text{x}^2\big)\frac{\text{d}}{\text{dx}}\big(\text{x}^2\big)+2\sin\text{x}\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}+2\sin^2\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}^2$
$=2\text{x}\cos\text{x}^2+2\cdot\sin\text{x}\cdot\cos\text{x}+2\sin\text{x}^2\cos\text{x}^2\cdot\frac{\text{d}}{\text{dx}}\text{x}^2$
$=2\text{x}\cos\text{x}^2+2\cdot\sin\text{x}\cdot\cos\text{x}+2\sin\text{x}^2\cos\text{x}^2\cdot2\text{x}$
$=2\text{x}\cos\text{x}^2+\sin2\text{x}+\sin\big(2\text{x}^2\big)\cdot2\text{x}$
$=2\text{x}\cos\text{x}^2+2\text{x}\cdot\sin2\big(\text{x}^2\big)+\sin2\text{x}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\sin\big(\text{x}^2\big)+\frac{\text{d}}{\text{dx}}\big(\sin\text{x}\big)^2+\frac{\text{d}}{\text{dx}}(\sin\text{x}^2)^2$
$=\cos\big(\text{x}^2\big)\frac{\text{d}}{\text{dx}}\big(\text{x}^2\big)+2\sin\text{x}\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}+2\sin^2\cdot\frac{\text{d}}{\text{dx}}\sin\text{x}^2$
$=2\text{x}\cos\text{x}^2+2\cdot\sin\text{x}\cdot\cos\text{x}+2\sin\text{x}^2\cos\text{x}^2\cdot\frac{\text{d}}{\text{dx}}\text{x}^2$
$=2\text{x}\cos\text{x}^2+2\cdot\sin\text{x}\cdot\cos\text{x}+2\sin\text{x}^2\cos\text{x}^2\cdot2\text{x}$
$=2\text{x}\cos\text{x}^2+\sin2\text{x}+\sin\big(2\text{x}^2\big)\cdot2\text{x}$
$=2\text{x}\cos\text{x}^2+2\text{x}\cdot\sin2\big(\text{x}^2\big)+\sin2\text{x}$
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