What is the derivative of ln(sqrt((1-sin(x)/(1+sin(x))))) ?

The derivative of ln(sqrt((1-sin(x)/(1+sin(x))))) is -sec(x)

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

The first derivative of ln(sqrt((1-sin(x)/(1+sin(x))))) is -sec(x)

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\frac{d}{d x} (ln (\frac{1 - sin ( x )}{1 + sin ( x )}))

- sec (x)

Solution steps

\frac{d}{d x} (ln (\frac{1 - sin ( x )}{1 + sin ( x )}))

Apply the chain rule:

\frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} \frac{d}{d x} (\frac{1 - sin ( x )}{1 + sin ( x )})

= \frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} \frac{d}{d x} (\frac{1 - sin ( x )}{1 + sin ( x )})

\frac{d}{d x} (\frac{1 - sin ( x )}{1 + sin ( x )}) = - \frac{cos ( x )}{- sin ( x ) + 1 ( sin ( x ) + 1 ) ^{\frac{3}{2}}}

= \frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} (- \frac{cos ( x )}{- sin ( x ) + 1 ( sin ( x ) + 1 ) ^{\frac{3}{2}}})

Simplify

\frac{1}{\frac{1 - s i n ( x )}{1 + s i n ( x )}} (- \frac{cos ( x )}{- sin ( x ) + 1 ( sin ( x ) + 1 ) ^{\frac{3}{2}}}) : - sec (x)

= - sec (x)

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