Derivative of xe^2csc(x)

1. Apply the product rule:

   $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

   $f{\left(x \right)} = e^{2} x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Apply the power rule: $x$ goes to $1$

      So, the result is: $e^{2}$

   $g{\left(x \right)} = \csc{\left(x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   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 result is: $- \frac{x e^{2} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + e^{2} \csc{\left(x \right)}$

2. Now simplify:

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

The answer is:

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