Derivative of Arccosine Function/Corollary
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Corollary to Derivative of Arccosine Function
Let $a \in \R$ be a constant
Let $x \in \R$ be a real number such that $x^2 < a^2$.
Let $\map \arccos {\dfrac x a}$ be the arccosine of $\dfrac x a$.
Then:
- $\map {\dfrac \d {\d x} } {\map \arccos {\dfrac x a} } = \dfrac {-1} {\sqrt {a^2 - x^2} }$
Proof
\(\ds \map {\dfrac \d {\d x} } {\map \arccos {\dfrac x a} }\) | \(=\) | \(\ds \frac 1 a \frac {-1} {\sqrt {1 - \paren {\frac x a}^2} }\) | Derivative of Arccosine 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
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $12$.