Derivative of Cosecant Function

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Theorem

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

where $\sin x \ne 0$.


Proof 1

From the definition of the cosecant function:

$\csc x = \dfrac 1 {\sin x}$

From Derivative of Sine Function:

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


Then:

\(\ds \map {\dfrac \d {\d x} } {\csc x}\) \(=\) \(\ds \cos x \frac {-1} {\sin^2 x}\) Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \frac {-1} {\sin x} \frac {\cos x} {\sin x}\)
\(\ds \) \(=\) \(\ds -\csc x \cot x\) Definition of Real Cosecant Function and Definition of Real Cotangent Function

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

$\blacksquare$


Proof 2

\(\ds \map {\dfrac \d {\d x} } {\csc x}\) \(=\) \(\ds \map {\dfrac \d {\d x} } {\dfrac 1 {\sin x} }\) Definition of Real Cosecant Function
\(\ds \) \(=\) \(\ds \dfrac {\sin x \map {\frac \d {\d x} } 1 - 1 \map {\frac \d {\d x} }{\sin x} } {\sin^2 x}\) Quotient Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {0 - \cos x} {\sin^2 x}\) Derivative of Sine Function, Derivative of Constant
\(\ds \) \(=\) \(\ds -\csc x \cot x\) Definition of Real Cosecant Function, Definition of Real Cotangent Function

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

$\blacksquare$


Also see


Sources

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