What is the derivative of ln|coth(5x|) ?

The derivative of ln|coth(5x|) is -5csch^2(5x)tanh(5x)

What is the first derivative of ln|coth(5x|) ?

The first derivative of ln|coth(5x|) is -5csch^2(5x)tanh(5x)

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\frac{d}{d x} (ln ∣ coth (5 x) ∣)

- 5 csch^{2} (5 x) tanh (5 x)

Solution steps

\frac{d}{d x} (ln ∣ coth (5 x) ∣)

Apply the chain rule:

\frac{1}{∣ coth ( 5 x ) ∣} \frac{d}{d x} (∣ coth (5 x) ∣)

= \frac{1}{∣ coth ( 5 x ) ∣} \frac{d}{d x} (∣ coth (5 x) ∣)

\frac{d}{d x} (∣ coth (5 x) ∣) = - \frac{5 csch ^{2} ( 5 x ) coth ( 5 x )}{∣ coth ( 5 x ) ∣}

= \frac{1}{∣ coth ( 5 x ) ∣} (- \frac{5 csch ^{2} ( 5 x ) coth ( 5 x )}{∣ coth ( 5 x ) ∣})

Simplify

\frac{1}{∣ coth ( 5 x ) ∣} (- \frac{5 csch ^{2} ( 5 x ) coth ( 5 x )}{∣ coth ( 5 x ) ∣}) : - 5 csch^{2} (5 x) tanh (5 x)

= - 5 csch^{2} (5 x) tanh (5 x)

What is the derivative of ln|coth(5x|) ?
The derivative of ln|coth(5x|) is -5csch^2(5x)tanh(5x)
What is the first derivative of ln|coth(5x|) ?
The first derivative of ln|coth(5x|) is -5csch^2(5x)tanh(5x)