The derivative of sinx+sinx with respective to x is

y=sinx+sinxy1=12sinx+sinx×[cosx+12sinx×cosx]=12sinx+sinx×[cosx[1+2sinx]2sinx]=cosx[1+2sinx]4sin2x+sinxsinx

The derivative of sinx+sinx with respective to x is

y=sinx+sinxy1=12sinx+sinx×[cosx+12sinx×cosx]=12sinx+sinx×[cosx[1+2sinx]2sinx]=cosx[1+2sinx]4sin2x+sinxsinx

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The derivative of $\sqrt{\sin x + \sqrt{\sin x}}$ with respective to x is

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$\frac{\cos x (1 + 2 \sqrt{\sin x})}{4 \sqrt{\sin^{2} x + \sin x \sqrt{\sin x}}}$

$\frac{\cos x (1 + 2 \sqrt{\sin x})}{4 \sqrt{\sin^{2} x - \sin x \sqrt{\sin x}}}$

$\frac{\cos x (1 - 2 \sqrt{\sin x})}{4 \sqrt{\sin^{2} x + \sin x \sqrt{\sin x}}}$

$\frac{\cos x (1 - 2 \sqrt{\sin x})}{4 \sqrt{\sin^{2} x - \sin x \sqrt{\sin x}}}$

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Detailed Solution

$y = \sqrt{\sin x + \sqrt{\sin x}}$

$y^{1} = \frac{1}{2 \sqrt{\sin x + \sqrt{\sin x}}} \times [\cos x + \frac{1}{2 \sqrt{\sin x}} \times \cos x]$

$= \frac{1}{2 \sqrt{\sin x + \sqrt{\sin x}}} \times [\frac{\cos x [1 + 2 \sqrt{\sin x}]}{2 \sqrt{\sin x}}]$

$= \frac{\cos x [1 + 2 \sqrt{\sin x}]}{4 \sqrt{\sin^{2} x + \sin x \sqrt{\sin x}}}$

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The derivative of sinx+sinx with respective to x is

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The derivative of $\sqrt{\sin x + \sqrt{\sin x}}$ with respective to x is