Derivative of xcosecx

1. Apply the product rule:

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

   $f{\left(x \right)} = c x \cos{\left(e \right)}$; 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. 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: $\cos{\left(e \right)}$

      So, the result is: $c \cos{\left(e \right)}$

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

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

   The result is: $c x \cos{\left(e \right)} + c x \cos{\left(e \right)}$

2. Now simplify:

   $$2 c x \cos{\left(e \right)}$$

The answer is:

$$2 c x \cos{\left(e \right)}$$