Derivative of Arcsine Function/Corollary

Corollary to Derivative of Arcsine Function

Let $a \in \R$ be a constant

Let $x \in \R$ be a real number such that $x^2 < a^2$.

Let $\map \arcsin {\dfrac x a}$ be the arcsine of $\dfrac x a$.

Then:

$\dfrac {\map \d {\map \arcsin {\frac x a} } } {\d x} = \dfrac 1 {\sqrt {a^2 - x^2} }$

Proof

\(\ds \frac {\map \d {\map \arcsin {\frac x a} } } {\d x}\)

\(=\)

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

Derivative of Arcsine Function and Derivative of Function of Constant Multiple

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

$\blacksquare$

Also see

• Derivative of $\arccos \dfrac x a$

• Derivative of $\arctan \dfrac x a$

• Derivative of $\arccot \dfrac x a$

• Derivative of $\arcsec \dfrac x a$

• Derivative of $\arccsc \dfrac x a$

Sources

• 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $12$.