What is the derivative of ln(tanh(x/2)) ?

The derivative of ln(tanh(x/2)) is (sech^2(x/2))/(2tanh(x/2))

What is the first derivative of ln(tanh(x/2)) ?

The first derivative of ln(tanh(x/2)) is (sech^2(x/2))/(2tanh(x/2))

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\frac{d}{d x} (ln (tanh (\frac{x}{2})))

\frac{sech ^{2} ( \frac{x}{2} )}{2 tanh ( \frac{x}{2} )}

Solution steps

\frac{d}{d x} (ln (tanh (\frac{x}{2})))

Apply the chain rule:

\frac{1}{tanh ( \frac{x}{2} )} \frac{d}{d x} (tanh (\frac{x}{2}))

= \frac{1}{tanh ( \frac{x}{2} )} \frac{d}{d x} (tanh (\frac{x}{2}))

\frac{d}{d x} (tanh (\frac{x}{2})) = sech^{2} (\frac{x}{2}) \frac{1}{2}

= \frac{1}{tanh ( \frac{x}{2} )} sech^{2} (\frac{x}{2}) \frac{1}{2}

Simplify

\frac{1}{tanh ( \frac{x}{2} )} sech^{2} (\frac{x}{2}) \frac{1}{2} : \frac{sech ^{2} ( \frac{x}{2} )}{2 tanh ( \frac{x}{2} )}

= \frac{sech ^{2} ( \frac{x}{2} )}{2 tanh ( \frac{x}{2} )}

What is the derivative of ln(tanh(x/2)) ?
The derivative of ln(tanh(x/2)) is (sech^2(x/2))/(2tanh(x/2))
What is the first derivative of ln(tanh(x/2)) ?
The first derivative of ln(tanh(x/2)) is (sech^2(x/2))/(2tanh(x/2))