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Calculus learning
Derivative of csc(x): Reciprocal Rule and Composite Functions
The derivative of csc(x) follows from csc(x)=1/sin(x). Keep both the minus sign from the reciprocal rule and the cosine from differentiating the inner sine.
Direct answer
Angles are in radians; sin(x) must not be zero.
Derivative of csc(x) worked method
Write the function as [sin(x)]⁻¹. The negative-power rule contributes −[sin(x)]⁻².
Multiply by the inner derivative cos(x). This gives −cos(x)/sin²(x).
Rewrite one denominator factor as csc and the other quotient as cot, giving −csc(x)cot(x).
For csc(2x), multiply by 2 and keep 2x in both trigonometric factors.
Worked examples
Not −csc²(x).
The factor 2 is outside.
Use the product rule as well.
Common mistakes
- • Confusing the derivative of csc with that of cot.
- • Losing the inner derivative.
- • Treating csc as inverse sine.
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.
Use 1/sin(x) in the existing engine. The lesson formula is equivalent on the stated domain.
Open the calculator without a preset: 微分References
Frequently asked questions
Is csc(x) the same as arcsin(x)?+
No. csc is a reciprocal, arcsin is an inverse function.
Why does the derivative contain cosine?+
The chain rule differentiates sin(x) inside the reciprocal.
Can I enter csc in the current preset?+
The preset uses 1/sin(x), avoiding an unsupported literal alias.
Conclusion
Derivative of csc(x) starts with the stated domain and conditions. Recheck those before carrying a worked example into a different problem.
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