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
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $13$.