What is the derivative of 2arctan(sqrt(sin(x))) ?

The derivative of 2arctan(sqrt(sin(x))) is (cos(x))/(sqrt(sin(x))(sin(x)+1))

What is the first derivative of 2arctan(sqrt(sin(x))) ?

The first derivative of 2arctan(sqrt(sin(x))) is (cos(x))/(sqrt(sin(x))(sin(x)+1))

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\frac{d}{d x} (2 arctan (sin (x)))

\frac{cos ( x )}{sin ( x ) ( sin ( x ) + 1 )}

Solution steps

\frac{d}{d x} (2 arctan (sin (x)))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{cos ( x )}{sin ( x ) ( sin ( x ) + 1 )}

What is the derivative of 2arctan(sqrt(sin(x))) ?
The derivative of 2arctan(sqrt(sin(x))) is (cos(x))/(sqrt(sin(x))(sin(x)+1))
What is the first derivative of 2arctan(sqrt(sin(x))) ?
The first derivative of 2arctan(sqrt(sin(x))) is (cos(x))/(sqrt(sin(x))(sin(x)+1))