What is the first (ln(tan(x)))^' ?

The first (ln(tan(x)))^' is 2csc(2x)

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(ln(tan(x)))^'

(ln (tan (x)))^{^{'}}

2 csc (2 x)

Solution steps

(ln (tan (x)))^{'}

Apply the chain rule:

\frac{1}{tan ( x )} (tan (x))^{'}

= \frac{1}{tan ( x )} (tan (x))^{'}

(tan (x))^{'} = sec^{2} (x)

= \frac{1}{tan ( x )} sec^{2} (x)

Simplify

\frac{1}{tan ( x )} sec^{2} (x) : 2 csc (2 x)

= 2 csc (2 x)