Derivative of Arctangent Function

Theorem

Let $x \in \R$.

Let $\arctan x$ be the arctangent of $x$.

Then:

$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {1 + x^2}$

Corollary

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

Proof 1

$\ds y$

$=$

$\ds \arctan x$

$\ds \leadsto \ \ $

$\ds x$

$=$

$\ds \tan y$

Definition of Real Arctangent

$\ds \leadsto \ \ $

$\ds \frac {\d x} {\d y}$

$=$

$\ds \sec^2 y$

Derivative of Tangent Function

$\ds $

$=$

$\ds 1 + \tan^2 y$

Difference of Squares of Secant and Tangent

$\ds $

$=$

$\ds 1 + x^2$

Definition of $x$

$\ds \leadsto \ \ $

$\ds \frac {\d y} {\d x}$

$=$

$\ds \frac 1 {1 + x^2}$

Derivative of Inverse Function

$\blacksquare$

Proof 2

$\ds \frac {\map \d {\arctan x} } {\d x}$

$=$

$\ds \lim_{h \mathop \to 0} \frac {\map \arctan {x + h} - \arctan x} h$

Definition of Derivative of Real Function at Point

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac {\map \arctan {x + h} + \map \arctan {-x} } h$

Arctangent Function is Odd

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac 1 h \map \arctan {\frac {x + h - x} {1 + x \paren {x + h} } }$

Sum of Arctangents

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac 1 h \map \arctan {\frac h {1 + x^2 + h x} }$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac 1 h \paren {\frac h {1 + x^2 + h x} - \frac 1 3 \paren {\frac h {1 + x^2 + h x} }^3 + \frac 1 5 \paren {\frac h {1 + x^2 + h x} }^5 + \map \OO {h^7} }$

Definition of Real Arctangent

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \paren {\frac 1 {1 + x^2 + h x} - \frac {h^2} {3 \paren {1 + x^2 + h x}^3} + \frac {h^4} {5 \paren {1 + x^2 + h x}^5} + \map \OO {h^6} }$

$\ds $

$=$

$\ds \frac 1 {1 + x^2 + 0 x} - \frac {0^2} {3 \paren {1 + x^2 + 0 x}^3} + \frac {0^4} {5 \paren {1 + x^2 + 0 x}^5}$

$\ds $

$=$

$\ds \frac 1 {1 + x^2}$

$\blacksquare$

Also presented as

The can also be presented in the form:

$\dfrac {\map \d {\arctan x} } {\d x} = \dfrac 1 {x^2 + 1}$

Also see

Sources

1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation
1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Derivatives of fundamental functions: $4.$ Inverse trigonometric functions
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.5 \ (4)$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arc-tangent
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: Inverse trigonometric functions
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse trigonometric function
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives