Derivative of csc(x)
The derivative of the cosecant is $\dfrac{d}{dx}\csc x = -\csc x\cot x = -\dfrac{\cos x}{\sin^2 x}$, for $x \neq k\pi$.
$$ \frac{d}{dx}\left(\csc x\right) = -\csc x\cot x $$
See the derivative of csc(x) step by step in the calculator →
📈 Graph of csc(x) and its derivative
🔍 Why is the derivative of csc(x) equal to $-\csc x\cot x$?
$\csc x = (\sin x)^{-1}$; by the chain rule
$$ (\csc x)' = -(\sin x)^{-2}\cos x = -\frac{1}{\sin x}\cdot\frac{\cos x}{\sin x} = -\csc x\cot x. $$
🔗 Derivative of csc(x) with an inner function
If the argument is a function $u(x)$, the chain rule adds the inner derivative:
$$ \frac{d}{dx}\csc\big(u(x)\big) = -u'(x)\csc\big(u(x)\big)\cot\big(u(x)\big) $$
✏️ Worked examples
⚠️ Common mistakes
Writing $+\csc x\cot x$ – the minus sign belongs to every "co"-function.
Confusing $\csc x$ with $\arcsin x = \sin^{-1}x$.
❓ Frequently asked questions
What is the derivative of 1/sin(x)?
$-\frac{\cos x}{\sin^2 x} = -\csc x\cot x$.
What is the derivative of csc²(x)?
$-2\csc^2 x\cot x$.
Where is the derivative of csc(x) zero?
Where $\cos x = 0$, i.e. at $x = \frac{\pi}{2} + k\pi$.