What is the derivative of ln(sqrt((3x+1/(x^2-2)))) ?

The derivative of ln(sqrt((3x+1/(x^2-2)))) is (-3x^2-2x-6)/(2(x^2-2)(3x+1))

What is the first derivative of ln(sqrt((3x+1/(x^2-2)))) ?

The first derivative of ln(sqrt((3x+1/(x^2-2)))) is (-3x^2-2x-6)/(2(x^2-2)(3x+1))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

3 x e^{x} csc (x)

x \to 0 lim (sin^{2} (x) (\frac{1}{x}))

x \to - 1 lim (\frac{1}{x ^{2} + 1})

n = 0 \sum \infty \frac{6}{5 ^{n}}

\int 2 sec (2 x) tan (2 x) d x

What is the derivative of ln(sqrt((3x+1/(x^2-2)))) ?
The derivative of ln(sqrt((3x+1/(x^2-2)))) is (-3x^2-2x-6)/(2(x^2-2)(3x+1))
What is the first derivative of ln(sqrt((3x+1/(x^2-2)))) ?
The first derivative of ln(sqrt((3x+1/(x^2-2)))) is (-3x^2-2x-6)/(2(x^2-2)(3x+1))