What is the derivative of $f(x)=arcsin sqrt sinx$ ?

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Answer 1 · 2015-07-03T12:29:25

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

$y = arcsinx$
$siny = x$
$cosy((dy)/(dx)) = 1$
$(dy)/(dx) = 1/(cosy) = 1/(sqrt(1-sin^2y)) = 1/(sqrt(1-x^2))$

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

Thus, take this further with the Chain Rule.

$d/(dx)[arcsinsqrt(sinx)] = 1/(sqrt(1-(sqrtsinx)^2)) * 1/((2sqrtsinx)) * cosx$

$= color(blue)(cosx/(2sqrtsinxsqrt(1-sinx)))$

What is the derivative of f(x)=arcsin sqrt sinxf(x)=arcsin sqrt sinx?