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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

x \to 0 + lim (\frac{x}{15 x ^{4} + 12 x})

x \to 2 lim (\frac{x ^{2} - 4}{x - 3})

\int (\frac{2 - x}{2})^{2} d x

6 t

\frac{d}{d x} (\frac{4}{1 - x})

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