What is (\partial)/(\partial x)(ln(sqrt(1+x^2))) ?

The answer to (\partial)/(\partial x)(ln(sqrt(1+x^2))) is x/(1+x^2)

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(\partial)/(\partial x)(ln(sqrt(1+x^2)))

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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What is (\partial)/(\partial x)(ln(sqrt(1+x^2))) ?
The answer to (\partial)/(\partial x)(ln(sqrt(1+x^2))) is x/(1+x^2)