What is the derivative of ln(sqrt(((x-3))/(x-1))) ?

The derivative of ln(sqrt(((x-3))/(x-1))) is 1/((x-1)(x-3))

What is the first derivative of ln(sqrt(((x-3))/(x-1))) ?

The first derivative of ln(sqrt(((x-3))/(x-1))) is 1/((x-1)(x-3))

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derivative

\frac{1}{( x - 1 ) ( x - 3 )}

Solution steps

\frac{d}{d x} (ln (\frac{x - 3}{x - 1}))

Apply the chain rule:

\frac{1}{\frac{x - 3}{x - 1}} \frac{d}{d x} (\frac{x - 3}{x - 1})

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

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

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

Simplify

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

= \frac{1}{( x - 1 ) ( x - 3 )}

What is the derivative of ln(sqrt(((x-3))/(x-1))) ?
The derivative of ln(sqrt(((x-3))/(x-1))) is 1/((x-1)(x-3))
What is the first derivative of ln(sqrt(((x-3))/(x-1))) ?
The first derivative of ln(sqrt(((x-3))/(x-1))) is 1/((x-1)(x-3))