What is the derivative of y=ln(sqrt((x+1)/(x-1))) ?

The derivative of y=ln(sqrt((x+1)/(x-1))) is -1/((x-1)(x+1))

What is the first derivative of y=ln(sqrt((x+1)/(x-1))) ?

The first derivative of y=ln(sqrt((x+1)/(x-1))) is -1/((x-1)(x+1))

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derivative

- \frac{1}{( x - 1 ) ( x + 1 )}

Solution steps

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

Apply the chain rule:

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

= \frac{1}{\frac{x + 1}{x - 1}} \frac{d}{d x} (\frac{x + 1}{x - 1})

\frac{d}{d x} (\frac{x + 1}{x - 1}) = - \frac{1}{x + 1 ( x - 1 ) ^{\frac{3}{2}}}

= \frac{1}{\frac{x + 1}{x - 1}} (- \frac{1}{x + 1 ( x - 1 ) ^{\frac{3}{2}}})

Simplify

\frac{1}{\frac{x + 1}{x - 1}} (- \frac{1}{x + 1 ( x - 1 ) ^{\frac{3}{2}}}) : - \frac{1}{( x - 1 ) ( x + 1 )}

= - \frac{1}{( x - 1 ) ( x + 1 )}

What is the derivative of y=ln(sqrt((x+1)/(x-1))) ?
The derivative of y=ln(sqrt((x+1)/(x-1))) is -1/((x-1)(x+1))
What is the first derivative of y=ln(sqrt((x+1)/(x-1))) ?
The first derivative of y=ln(sqrt((x+1)/(x-1))) is -1/((x-1)(x+1))