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

The derivative of ln((1+sin(x)/(1-sin(x)))) is 2sec(x)

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

The first derivative of ln((1+sin(x)/(1-sin(x)))) is 2sec(x)

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

2 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{2 cos ( x )}{( 1 - sin ( x ) ) ^{2}}

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

Simplify

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

= 2 sec (x)

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