Derivative of Real Area Hyperbolic Tangent of x over a

Theorem

$\map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} } = \dfrac a {a^2 - x^2}$

where $-a < x < a$.

Corollary

$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } } = \dfrac 1 {a^2 - x^2}$

where $\size x < a$.

Proof

Let $-a < x < a$.

Then $-1 < \dfrac x a < 1$ and so:

\(\ds \map {\dfrac \d {\d x} } {\map \artanh {\dfrac x a} }\)

\(=\)

\(\ds \frac 1 a \frac 1 {1 - \paren {\frac x a}^2}\)

Derivative of $\artanh$ and Derivative of Function of Constant Multiple

\(\ds \)

\(=\)

\(\ds \frac 1 a \frac 1 {\frac {a^2 - x^2} {a^2} }\)

\(\ds \)

\(=\)

\(\ds \frac 1 a \frac {a^2} {a^2 - x^2}\)

\(\ds \)

\(=\)

\(\ds \frac a {a^2 - x^2}\)

$\artanh \dfrac x a$ is not defined when either $x \le -a$ or $x \ge a$.

$\blacksquare$

Also presented as

Some sources present this as:

$\map {\dfrac \d {\d x} } {\dfrac 1 a \map \artanh {\dfrac x a} } = \dfrac 1 {a^2 - x^2}$

Also see

• Derivative of $\arsinh \dfrac x a$

• Derivative of $\arcosh \dfrac x a$

• 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: $16$.