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Calculus learning

Derivative of cot(x): Quotient Rule, Sign and Chain Rule

The derivative of cot(x) is negative because cot is cos divided by sin. Deriving the formula with the quotient rule also explains its denominator and the points where the function is undefined.

Direct answer

d/dx cot(x)=−csc²(x). For cot(g(x)), multiply by g′(x).

Use radians and exclude sin(x)=0. For a composition, exclude sin(g(x))=0.

Derivative of cot(x) worked method

Rewrite cot(x)=cos(x)/sin(x) on its original domain.

The quotient numerator is (−sin x)sin x−cos x(cos x)=−(sin²x+cos²x)=−1.

Divide by sin²x to obtain −1/sin²x=−csc²x.

For cot(3x), the inner derivative is 3, giving −3csc²(3x). The excluded points are 3x=kπ.

1
cot x=cos x/sin x
2
(cot x)′=−1/sin²x
3
(cot(g(x)))′=−g′(x)/sin²(g(x))

Worked examples

d/dx cot(x)
−csc²x

Negative on every domain interval.

d/dx cot(3x)
−3csc²(3x)

Multiply by 3.

d/dx cot(x²)
−2x csc²(x²)

Keep the original inner expression.

Common mistakes

  • • Copying the positive sign from the derivative of tan.
  • • Replacing cot by tan.
  • • Ignoring the excluded poles.

Try the supported calculator preset

The preset opens a supported expression with its variable and conditions. Teaching examples elsewhere in this guide are reference explanations, not claims that the engine supports every method.

The input intentionally uses the quotient representation supported by the current parser.

微分:cos⁡xsin⁡x\frac{\cos x}{\sin x}
Open the calculator without a preset: 微分

References

Frequently asked questions

Is cot the inverse of tan?+

No. cot=1/tan where both expressions are defined; arctan is an inverse function.

Can I use degrees directly?+

The standard formula assumes radians. Degree arguments introduce a π/180 factor.

What input does the preset use?+

cos(x)/sin(x), the supported equivalent notation. It does not claim a new cot parser alias.

Conclusion

Derivative of cot(x) starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.

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