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.