To differentiate the cosecant function (\(\csc(x)\)) with respect to \(x\), we use the following derivative formula:

\[
\frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x)
\]

So, the derivative of \(\csc(x)\) with respect to \(x\) is:

\[
\frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x)
\]

This result comes from using the chain rule and the basic trigonometric identities.

Would you like more details on the derivation? Here are some related questions to explore:

1. What is the derivative of \(\sec(x)\)?
2. How is the derivative of \(\cot(x)\) derived?
3. What are the basic trigonometric identities used in differentiation?
4. How do you apply the chain rule in trigonometric differentiation?
5. What is the significance of the negative sign in the derivative of \(\csc(x)\)?

**Tip:** When differentiating trigonometric functions, it's helpful to remember their basic identities and derivatives.

Learn how to differentiate the cosecant function csc(x) with a step-by-step guide. This problem involves trigonometric differentiation using the formula \( \frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x) \), along with basic trigonometric identities and the chain rule. Suitable for advanced high school students and those studying calculus.

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