What is the derivative of ln|tan(x|) ?

The derivative of ln|tan(x|) is 2csc(2x)

What is the first derivative of ln|tan(x|) ?

The first derivative of ln|tan(x|) is 2csc(2x)

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\frac{d}{d x} (ln ∣ tan (x) ∣)

2 csc (2 x)

Solution steps

\frac{d}{d x} (ln ∣ tan (x) ∣)

Apply the chain rule:

\frac{1}{∣ tan ( x ) ∣} \frac{d}{d x} (∣ tan (x) ∣)

= \frac{1}{∣ tan ( x ) ∣} \frac{d}{d x} (∣ tan (x) ∣)

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

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

Simplify

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

= 2 csc (2 x)

What is the derivative of ln|tan(x|) ?
The derivative of ln|tan(x|) is 2csc(2x)
What is the first derivative of ln|tan(x|) ?
The first derivative of ln|tan(x|) is 2csc(2x)