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

From ProofWiki
Jump to navigation Jump to search

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
\(\ds \leadsto \ \ \) \(\ds \map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 - a^2} } }\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\map \arcosh {\frac x a} - \ln a }\)
\(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt {x^2 - a^2} } + \map {\dfrac \d {\d x} } {\ln a}\) Derivative of Real Area Hyperbolic 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

Retrieved from "https://proofwiki.org/w/index.php?title=Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a/Corollary_1&oldid=513687"