To derive the derivative of \( \csc(x) \), we follow these steps:

### Step-by-step process:

1. **Rewrite the function using its definition:**
   \[
   \csc(x) = \frac{1}{\sin(x)}
   \]
   This makes the differentiation process easier since we can apply the quotient rule.

2. **Differentiate using the quotient rule:**
   The quotient rule states that for a function \( f(x) = \frac{g(x)}{h(x)} \), the derivative is given by:
   \[
   \frac{d}{dx}\left[\frac{g(x)}{h(x)}\right] = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2}
   \]
   In this case, \( g(x) = 1 \) and \( h(x) = \sin(x) \), so we get:
   \[
   \frac{d}{dx} \left( \csc(x) \right) = \frac{(0)\sin(x) - (1)\cos(x)}{(\sin(x))^2}
   \]
   Simplifying this expression gives:
   \[
   \frac{d}{dx} \left( \csc(x) \right) = -\frac{\cos(x)}{\sin^2(x)}
   \]

3. **Rewrite the result in terms of trigonometric identities:**
   \[
   \frac{\cos(x)}{\sin^2(x)} = \csc(x)\cot(x)
   \]
   Therefore, the derivative of \( \csc(x) \) is:
   \[
   \frac{d}{dx} \left( \csc(x) \right) = -\csc(x)\cot(x)
   \]

### Final Answer:
\[
\frac{d}{dx} \left( \csc(x) \right) = -\csc(x)\cot(x)
\]

Would you like more details on any step or further examples?

### Related Questions:
1. What is the derivative of \( \sec(x) \)?
2. How do you derive the derivative of \( \cot(x) \)?
3. Can you prove the quotient rule?
4. How do you find the derivative of a composite trigonometric function?
5. What are the second derivatives of trigonometric functions?

### Tip:
When differentiating trigonometric functions, try rewriting them using basic identities to simplify the process.

Learn how to derive the derivative of csc(x) step by step using the quotient rule and trigonometric identities. This guide covers core calculus and trigonometry concepts suitable for students in Grades 11-12 or at the university level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation