Derivative of cosec(x)

1. Rewrite the function to be differentiated:

   $$\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}$$

2. Let $u = \sin{\left(x \right)}$.

3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$:

   1. The derivative of sine is cosine:

      $$\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$$

   The result of the chain rule is:

   $$- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

The answer is:

$$- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$