Derivative of Arctangent Function/Corollary

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Corollary to Derivative of Arctangent Function

Let $x \in \R$.

Let $\map \arctan {\dfrac x a}$ denote the arctangent of $\dfrac x a$.


Then:

$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {a^2 + x^2}$


Proof

\(\ds \frac {\map \d {\map \arctan {\frac x a} } } {\d x}\) \(=\) \(\ds \frac 1 a \frac 1 {1 + \paren {\frac x a}^2}\) Derivative of Arctangent 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:

$\dfrac {\map \d {\map \arctan {\frac x a} } } {\d x} = \dfrac a {x^2 + a^2}$


Also see


Sources