What is the derivative of arcsin(sqrt(sin(x))) ?

The derivative of arcsin(sqrt(sin(x))) is (cos(x))/(2sqrt(sin(x))\sqrt{1-sin(x))}

What is the first derivative of arcsin(sqrt(sin(x))) ?

The first derivative of arcsin(sqrt(sin(x))) is (cos(x))/(2sqrt(sin(x))\sqrt{1-sin(x))}

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\frac{d}{d x} (arcsin (sin (x)))

\frac{cos ( x )}{2 sin ( x ) 1 - sin ( x )}

Solution steps

\frac{d}{d x} (arcsin (sin (x)))

Apply the chain rule:

\frac{1}{1 - ( sin ( x ) ) ^{2}} \frac{d}{d x} (sin (x))

= \frac{1}{1 - ( sin ( x ) ) ^{2}} \frac{d}{d x} (sin (x))

\frac{d}{d x} (sin (x)) = \frac{cos ( x )}{2 sin ( x )}

= \frac{1}{1 - ( sin ( x ) ) ^{2}} \cdot \frac{cos ( x )}{2 sin ( x )}

Simplify

\frac{1}{1 - ( sin ( x ) ) ^{2}} \cdot \frac{cos ( x )}{2 sin ( x )} : \frac{cos ( x )}{2 sin ( x ) 1 - sin ( x )}

= \frac{cos ( x )}{2 sin ( x ) 1 - sin ( x )}

What is the derivative of arcsin(sqrt(sin(x))) ?
The derivative of arcsin(sqrt(sin(x))) is (cos(x))/(2sqrt(sin(x))\sqrt{1-sin(x))}
What is the first derivative of arcsin(sqrt(sin(x))) ?
The first derivative of arcsin(sqrt(sin(x))) is (cos(x))/(2sqrt(sin(x))\sqrt{1-sin(x))}