Derivative of Real Area Hyperbolic Cosine of x over a
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Theorem
- $\dfrac {\map \d {\map \arcosh {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {x^2 - a^2} }$
where $x > a$.
Corollary 1
- $\map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 - a^2} } } = \dfrac 1 {\sqrt {x^2 - a^2} }$
for $x > a$.
Corollary 2
- $\map {\dfrac \d {\d x} } {\map \ln {x - \sqrt {x^2 - a^2} } } = -\dfrac 1 {\sqrt {x^2 - a^2} }$
for $x > a$.
Proof
Let $x > a$.
Then $\dfrac x a > 1$ and so:
\(\ds \frac {\map \d {\map {\cosh^{-1} } {\frac x a} } } {\d x}\) | \(=\) | \(\ds \frac 1 a \frac 1 {\sqrt {\paren {\frac x a}^2} - 1}\) | Derivative of $\arcosh$ and Derivative of Function of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac 1 {\sqrt {\frac {x^2 - a^2} {a^2} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac a {\sqrt {x^2 - a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt {x^2 - a^2} }\) |
$\cosh^{-1} \dfrac x a$ is not defined when $x \le a$.
When $x = a$ we have that $\sqrt {x^2 - a^2} = 0$ and so $\dfrac 1 {\sqrt {x^2 - a^2} }$ is not defined.
Hence the restriction on the domain.
$\blacksquare$
Also see
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $15$.