Derivative of Real Area Hyperbolic Sine of x over a/Corollary 1

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Theorem

$\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$


Proof

\(\ds \map \arsinh {\frac x a}\) \(=\) \(\ds \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 + a^2} }\) Definition of Real Area Hyperbolic Sine
\(\ds \) \(=\) \(\ds \map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 + a^2} }\)
\(\ds \) \(=\) \(\ds \map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 + a^2} } }\)
\(\ds \) \(=\) \(\ds \map \ln {x + \sqrt {x^2 + a^2} } - \ln a\) Difference of Logarithms
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 + a^2} } }\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\map \arsinh {\frac x a} + \ln a }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt {x^2 + a^2} } + \map {\dfrac \d {\d x} } {\ln a}\) Derivative of Real Area Hyperbolic Sine of x over a
\(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt {x^2 + a^2} } + 0\) Derivative of Constant


We have that $\sqrt {x^2 + a^2} > x$ for all $x$.

Thus:

\(\ds x + \sqrt {x^2 + a^2}\) \(>\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \size {x + \sqrt {x^2 + a^2} }\) \(=\) \(\ds x + \sqrt {x^2 + a^2}\) Definition of Absolute Value
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\size {x + \sqrt {x^2 + a^2} } }\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 + a^2} } }\)

and the result follows.

$\blacksquare$


Sources

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