Derivative of cscx-sec(2x)

1. Differentiate $\csc{\left(x \right)} - \sec{\left(2 x \right)}$ term by term:

   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:

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

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

      1. Rewrite the function to be differentiated:

         $$\sec{\left(2 x \right)} = \frac{1}{\cos{\left(2 x \right)}}$$

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

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

      4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(2 x \right)}$:

         1. Let $u = 2 x$.

         2. The derivative of cosine is negative sine:

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$:

            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: $2$

            The result of the chain rule is:

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

         The result of the chain rule is:

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

      So, the result is: $- \frac{2 \sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$

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

2. Now simplify:

   $$\frac{\left(- 4 \sin^{2}{\left(x \right)} - 4 \sin{\left(x \right)} + 4 - \frac{1}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\left(2 \sin^{2}{\left(x \right)} - 1\right)^{2}}$$

The answer is: