Derivative of Cotangent Function

Theorem

$\map {\dfrac \d {\d x} } {\cot x} = -\csc^2 x = \dfrac {-1} {\sin^2 x}$

where $\sin x \ne 0$.

$\map {\dfrac \d {\d x} } {\cot a x} = -a \csc^2 a x$

$\dfrac \d {\d x} \cot x = -1 - \cot^2 x$

$\map {\dfrac \d {\d x} } {\cot a x} = -a \paren {\cot^2 a x + 1}$

From the definition of the cotangent function:

$\cot x = \dfrac {\cos x} {\sin x}$

$\map {\dfrac \d {\d x} } {\sin x} = \cos x$

$\map {\dfrac \d {\d x} } {\cos x}= -\sin x$

Then:

$\ds \map {\dfrac \d {\d x} } {\cot x}$

$=$

$\ds \frac {\sin x \paren {-\sin x} - \cos x \cos x} {\sin^2 x}$

$\ds $

$=$

$\ds \frac {-\paren {\sin^2 x + \cos^2 x} } {\sin^2 x}$

$\ds $

$=$

$\ds \frac {-1} {\sin^2 x}$

This is valid only when $\sin x \ne 0$.

The result follows from the definition of the real cosecant function.

$\blacksquare$

1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $7$.
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: $3.$ Trigonometric functions
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions: 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): Appendix $6$: Derivatives
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives