Derivative Calculator LogoDerivative Calculator

Derivative of Cosecant

Complete guide with formula, proof, examples, and graph.

Quick Answer

ddxcsc⁡(x)=−csc⁡(x)cot⁡(x)\frac{d}{dx}\csc(x) = -\csc(x)\cot(x)

The derivative of \csc(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Rewrite cosecant as a power of sine.

csc⁡(x)=1sin⁡(x)=[sin⁡(x)]−1\csc(x) = \frac{1}{\sin(x)} = [\sin(x)]^{-1}

Apply the chain rule with the power rule.

Chainrule:ddx[u]−1=−u−2⋅u′Chain rule: \frac{d}{dx}[u]^{-1} = -u^{-2} \cdot u'

Differentiate and multiply by the derivative of sin(x).

ddxcsc⁡(x)=−[sin⁡(x)]−2⋅cos⁡(x)\frac{d}{dx}\csc(x) = -[\sin(x)]^{-2} \cdot \cos(x)

Factor into −csc(x)cot(x).

=−cos⁡(x)sin⁡2(x)=−1sin⁡(x)⋅cos⁡(x)sin⁡(x)=−csc⁡(x)cot⁡(x)= \frac{-\cos(x)}{\sin^2(x)} = -\frac{1}{\sin(x)} \cdot \frac{\cos(x)}{\sin(x)} = -\csc(x)\cot(x)

Graph

Visualization of Cosecant and its derivative.

f(x) = \csc(x)

f(x)=csc⁡(x)f(x) = \csc(x)

f'(x) = -\csc(x)\cot(x)

f′(x)=−csc⁡(x)cot⁡(x)f'(x) = -\csc(x)\cot(x)
Domain: x≠kπ,k∈Zx \neq k\pi, k \in \mathbb{Z}Range: (−∞,−1]∪[1,+∞)(-\infty, -1] \cup [1, +\infty)

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: ddxcsc⁡(x3)\frac{d}{dx}\csc(x^3)

Solution: −3x2csc⁡(x3)cot⁡(x3)-3x^2\csc(x^3)\cot(x^3)

1.u=x3,u′=3x2u = x^3, u' = 3x^2
2.=−3x2csc⁡(x3)cot⁡(x3)= -3x^2\csc(x^3)\cot(x^3)

Calculate Any Derivative

Use our free online derivative calculator to verify your answers or solve more complex functions.

Open Derivative Calculator

Related Derivatives

sin⁡(x)\sin(x)cos⁡(x)\cos(x)tan⁡(x)\tan(x)cot⁡(x)\cot(x)sec⁡(x)\sec(x)ln⁡(x)\ln(x)log⁡a(x)\log_a(x)x\sqrt{x}arcsin⁡(x)\arcsin(x)

Frequently Asked Questions

What is the derivative of Cosecant?

+
The derivative of csc⁡(x)\csc(x) is −csc⁡(x)cot⁡(x)-\csc(x)\cot(x). This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Cosecant?

+
The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Cosecant always the same?

+
Yes, the derivative formula for csc⁡(x)\csc(x) is constant — it does not depend on x. However, when composed with inner functions (e.g., csc⁡(x)\csc(x) of u(x)), the chain rule applies.

Where is the derivative of Cosecant undefined?

+
The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Cosecant important?

+
This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.