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
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $14$.