# How do I find the derivative of F(x)=arcsin(sqrtsinx)?

Oct 27, 2015

$F ' \left(x\right) = \cos \frac{x}{2 \sqrt{\sin x} \sqrt{1 - \sin x}}$

#### Explanation:

$F \left(x\right) = f \left(g \left(x\right)\right) \implies F ' \left(x\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

$F ' \left(x\right) = \frac{1}{\sqrt{1 - {\left(\sqrt{\sin x}\right)}^{2}}} \cdot \left(\sqrt{\sin x}\right) '$

$F ' \left(x\right) = \frac{1}{\sqrt{1 - {\left(\sqrt{\sin x}\right)}^{2}}} \cdot \frac{1}{2 \sqrt{\sin x}} \cdot \left(\sin x\right) '$

$F ' \left(x\right) = \frac{1}{\sqrt{1 - \sin x}} \cdot \frac{1}{2 \sqrt{\sin x}} \cdot \cos x$

$F ' \left(x\right) = \cos \frac{x}{2 \sqrt{\sin x} \sqrt{1 - \sin x}}$