Derivative of Real Area Hyperbolic Sine of x over a
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Theorem
- $\dfrac {\map \d {\map \arsinh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 + a^2}}$
Corollary 1
- $\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$
Corollary 2
- $\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$
Proof
\(\ds \frac {\map \d {\map \arsinh {\frac x a} } } {\d x}\) | \(=\) | \(\ds \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} + 1}\) | Derivative of $\arsinh$ and Derivative of Function of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac 1 {\sqrt {\frac {x^2 + a^2} {a^2} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac a {\sqrt {x^2 + a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt {x^2 + a^2} }\) |
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $14$.