What is the derivative of ln(sqrt((x+6/(x-6)))) ?

The derivative of ln(sqrt((x+6/(x-6)))) is -6/((x-6)(x+6))

What is the first derivative of ln(sqrt((x+6/(x-6)))) ?

The first derivative of ln(sqrt((x+6/(x-6)))) is -6/((x-6)(x+6))

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\frac{d}{d x} (ln (\frac{x + 6}{x - 6}))

- \frac{6}{( x - 6 ) ( x + 6 )}

Solution steps

\frac{d}{d x} (ln (\frac{x + 6}{x - 6}))

Apply the chain rule:

\frac{1}{\frac{x + 6}{x - 6}} \frac{d}{d x} (\frac{x + 6}{x - 6})

= \frac{1}{\frac{x + 6}{x - 6}} \frac{d}{d x} (\frac{x + 6}{x - 6})

\frac{d}{d x} (\frac{x + 6}{x - 6}) = - \frac{6}{x + 6 ( x - 6 ) ^{\frac{3}{2}}}

= \frac{1}{\frac{x + 6}{x - 6}} (- \frac{6}{x + 6 ( x - 6 ) ^{\frac{3}{2}}})

Simplify

\frac{1}{\frac{x + 6}{x - 6}} (- \frac{6}{x + 6 ( x - 6 ) ^{\frac{3}{2}}}) : - \frac{6}{( x - 6 ) ( x + 6 )}

= - \frac{6}{( x - 6 ) ( x + 6 )}

y^{^{'}} = - \frac{2 x y}{1 + x ^{2}}

f (z) = \frac{1}{z}

\int 25 t^{4} e^{- t^{5}} d t

x \to 13 lim (π)

\frac{d}{d x} (10 x^{\frac{3}{2}})

What is the derivative of ln(sqrt((x+6/(x-6)))) ?
The derivative of ln(sqrt((x+6/(x-6)))) is -6/((x-6)(x+6))
What is the first derivative of ln(sqrt((x+6/(x-6)))) ?
The first derivative of ln(sqrt((x+6/(x-6)))) is -6/((x-6)(x+6))