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

The derivative of ln((sqrt(2x)/(sin(x)))) is (1-2xcot(x))/(2x)

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

The first derivative of ln((sqrt(2x)/(sin(x)))) is (1-2xcot(x))/(2x)

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

\frac{1 - 2 x cot ( x )}{2 x}

Solution steps

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

Simplify

ln (\frac{2 x}{sin ( x )}) : ln (2 x csc (x))

= \frac{d}{d x} (ln (2 x csc (x)))

Apply the chain rule:

\frac{1}{2 x csc ( x )} \frac{d}{d x} (2 x csc (x))

= \frac{1}{2 x csc ( x )} \frac{d}{d x} (2 x csc (x))

\frac{d}{d x} (2 x csc (x)) = 2 (\frac{csc ( x )}{2 x} - x cot (x) csc (x))

= \frac{1}{2 x csc ( x )} 2 (\frac{csc ( x )}{2 x} - x cot (x) csc (x))

Simplify

\frac{1}{2 x csc ( x )} 2 (\frac{csc ( x )}{2 x} - x cot (x) csc (x)) : \frac{1 - 2 x cot ( x )}{2 x}

= \frac{1 - 2 x cot ( x )}{2 x}

y = \frac{3}{sin ( 2 x )}

x \to 1 lim (\frac{( x + 3 )}{2})

\int (a x + b)^{n} d x

\int 27 - 6 x - x^{2} d x

\frac{d}{d x} (e^{3 x} \cdot cos (2 x))

What is the derivative of ln((sqrt(2x)/(sin(x)))) ?
The derivative of ln((sqrt(2x)/(sin(x)))) is (1-2xcot(x))/(2x)
What is the first derivative of ln((sqrt(2x)/(sin(x)))) ?
The first derivative of ln((sqrt(2x)/(sin(x)))) is (1-2xcot(x))/(2x)