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

The derivative of ln(sqrt(sin(x))) is (cot(x))/2

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

The first derivative of ln(sqrt(sin(x))) is (cot(x))/2

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

\frac{cot ( x )}{2}

Solution steps

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

Apply the chain rule:

\frac{1}{sin ( x )} \frac{d}{d x} (sin (x))

= \frac{1}{sin ( x )} \frac{d}{d x} (sin (x))

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

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

Simplify

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

= \frac{cot ( x )}{2}

What is the derivative of ln(sqrt(sin(x))) ?
The derivative of ln(sqrt(sin(x))) is (cot(x))/2
What is the first derivative of ln(sqrt(sin(x))) ?
The first derivative of ln(sqrt(sin(x))) is (cot(x))/2