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Popular >

derivative of ln((x^2/(1+x^2)))

Solution

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

Solution

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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x \to 0 lim (\frac{1}{2} + \frac{x}{2 ∣ x ∣})

Frequently Asked Questions (FAQ)

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