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

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

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

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

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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