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Derivative of Cotangent

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

Quick Answer

ddxcot⁡(x)=−csc⁡2(x)\frac{d}{dx}\cot(x) = -\csc^2(x)

The derivative of \cot(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Express cotangent as the quotient of cosine and sine.

cot⁡(x)=cos⁡(x)sin⁡(x)\cot(x) = \frac{\cos(x)}{\sin(x)}

Apply the quotient rule for differentiation.

Quotientrule:(uv)′=u′v−uv′v2Quotient rule: \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Substitute u = cos(x), v = sin(x), u' = −sin(x), v' = cos(x).

ddxcot⁡(x)=−sin⁡(x)⋅sin⁡(x)−cos⁡(x)⋅cos⁡(x)sin⁡2(x)\frac{d}{dx}\cot(x) = \frac{-\sin(x)\cdot\sin(x) - \cos(x)\cdot\cos(x)}{\sin^2(x)}

Simplify using the Pythagorean identity.

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

Graph

Visualization of Cotangent and its derivative.

f(x) = \cot(x)

f(x)=cot⁡(x)f(x) = \cot(x)

f'(x) = -\csc^2(x)

f′(x)=−csc⁡2(x)f'(x) = -\csc^2(x)
Domain: x≠kπ,k∈Zx \neq k\pi, k \in \mathbb{Z}Range: (−∞,+∞)(-\infty, +\infty)

Worked Examples

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

Find: ddxcot⁡(x2)\frac{d}{dx}\cot(x^2)

Solution: −2xcsc⁡2(x2)-2x\csc^2(x^2)

1.Chainrule:u=x2,u′=2xChain rule: u = x^2, u' = 2x
2.=−csc⁡2(x2)⋅2x= -\csc^2(x^2) \cdot 2x

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Related Derivatives

sin⁡(x)\sin(x)cos⁡(x)\cos(x)tan⁡(x)\tan(x)sec⁡(x)\sec(x)csc⁡(x)\csc(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 Cotangent?

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The derivative of cot⁡(x)\cot(x) is −csc⁡2(x)-\csc^2(x). This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Cotangent?

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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 Cotangent always the same?

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

Where is the derivative of Cotangent undefined?

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

Why is the derivative of Cotangent important?

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