Derivative of Arccotangent Function/Corollary
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Corollary to Derivative of Arccotangent Function
Let $x \in \R$.
Let $\arccot \dfrac x a$ be the arccotangent of $\dfrac x a$.
Then:
- $\dfrac {\map \d {\arccot \frac x a} } {\d x} = \dfrac {-a} {a^2 + x^2}$
Proof
\(\ds \frac {\map \d {\arccot \frac x a} } {\d x}\) | \(=\) | \(\ds \frac 1 a \frac {-1} {1 + \paren {\frac x a}^2}\) | Derivative of Arccotangent Function and Derivative of Function of Constant Multiple | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {-1} {\frac {a^2 + x^2} {a^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {-a^2} {a^2 + x^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-a} {a^2 + x^2}\) |
$\blacksquare$
Also defined as
This result can also be reported as:
- $\dfrac {\map \d {\arccot \frac x a} } {\d x} = \dfrac {-a} {x^2 + a^2}$