What is the derivative of $f (x) = arcsin \sqrt{sin x}$ $f(x)=arcsin sqrt sinx$ ?

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What is the derivative of $f (x) = arcsin \sqrt{sin x}$ $f(x)=arcsin sqrt sinx$ ?

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Answer 1 · 2015-07-03T12:29:25

One can derive the derivative for $arcsin x$ $arcsinx$ with implicit differentiation if it is not easy to remember it.

$y = arcsin x$ $y = arcsinx$
$sin y = x$ $siny = x$
$cos y (\frac{d y}{d x}) = 1$ $cosy((dy)/(dx)) = 1$
$\frac{d y}{d x} = \frac{1}{cos y} = \frac{1}{\sqrt{1 - {sin}^{2} y}} = \frac{1}{\sqrt{1 - x^{2}}}$ $(dy)/(dx) = 1/(cosy) = 1/(sqrt(1-sin^2y)) = 1/(sqrt(1-x^2))$

since ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x + cos^2x = 1$ .

Thus, take this further with the Chain Rule.

$\frac{d}{d x} [arcsin \sqrt{sin x}] = \frac{1}{\sqrt{1 - {(\sqrt{sin x})}^{2}}} \cdot \frac{1}{(2 \sqrt{sin x})} \cdot cos x$ $d/(dx)[arcsinsqrt(sinx)] = 1/(sqrt(1-(sqrtsinx)^2)) * 1/((2sqrtsinx)) * cosx$

$= \frac{cos x}{2 \sqrt{sin x} \sqrt{1 - sin x}}$ $= color(blue)(cosx/(2sqrtsinxsqrt(1-sinx)))$

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What is the derivative of $f (x) = arcsin \sqrt{sin x}$ $f(x)=arcsin sqrt sinx$ ?

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