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

d/dx csc(x)=−csc(x)cot(x)=−cos(x)/sin²(x).

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.

1
csc x=[sin x]⁻¹
2
(csc x)′=−cos x/sin²x
3
(csc(g(x)))′=−g′(x) cos(g(x))/sin²(g(x))

Worked examples

d/dx csc(x)
−csc(x)cot(x)

Not −csc²(x).

d/dx csc(2x)
−2csc(2x)cot(2x)

The factor 2 is outside.

d/dx [x csc(x)]
csc(x)−x csc(x)cot(x)

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.

Derivatives:1sin⁡x\frac{1}{\sin x}
Open the calculator without a preset: Derivatives

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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