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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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