Derivative of Real Area Hyperbolic Cosine of x over a/Corollary 2

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Theorem

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

for $x > a$.


Proof

\(\ds -\map {\arcosh} {\frac x a}\) \(=\) \(\ds -\map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - a^2} }\) Definition of Real Area Hyperbolic Cosine
\(\ds \) \(=\) \(\ds -\map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 - a^2} }\)
\(\ds \) \(=\) \(\ds -\map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 - a^2} } }\)
\(\ds \) \(=\) \(\ds -\map \ln {x + \sqrt {x^2 - a^2} } + \ln a\) Difference of Logarithms
\(\ds \) \(=\) \(\ds \map \ln {x - \sqrt {x^2 - a^2} } + \ln a^2 + \ln a\) Negative of Logarithm of x plus Root x squared minus a squared
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\map \ln {x - \sqrt {x^2 - a^2} } }\) \(=\) \(\ds \map {\dfrac \d {\d x} } {-\map {\cosh^{-1} } {\frac x a} - \ln a^3}\) Sum of Logarithms
\(\ds \) \(=\) \(\ds -\dfrac 1 {\sqrt {x^2 - a^2} } - \map {\dfrac \d {\d x} } {\ln a^3}\) Derivative of Real Area Hyperbolic Cosine of x over a
\(\ds \) \(=\) \(\ds -\dfrac 1 {\sqrt {x^2 - a^2} } + 0\) Derivative of Constant


When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and then $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.

When $\size x < a$ we have that $x^2 - a^2 < 0$ and then $\sqrt {x^2 - a^2}$ is not defined.

When $x < -a$ we have that $x - \sqrt {x^2 - a^2} < 0$ and so $\map \ln {x + \sqrt {x^2 - a^2} }$ is not defined.

Hence the restriction on the domain.

$\blacksquare$


Sources

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