Derivative of Arctangent Function/Corollary

Corollary to Derivative of Arctangent Function

Let $x \in \R$.

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

Then:

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

Proof

\(\ds \map {\dfrac \d {\d x} } {\map \arctan {\dfrac x a} }\)

\(=\)

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

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 presented as

This result can also be presented as:

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

Also see

• Derivative of $\arcsin \dfrac x a$

• Derivative of $\arccos \dfrac x a$

• 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: $13$.