Derivative of Real Area Hyperbolic Sine of x over a

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

• Derivative of $\arcosh \dfrac x a$

• Derivative of $\artanh \dfrac x a$

• Derivative of $\arcoth \dfrac x a$

• Derivative of $\arsech \dfrac x a$

• Derivative of $\arcsch \dfrac x a$

Sources

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