Derivative of csc(x)-(1/3)csc^3x

1. Differentiate $- \frac{\csc^{3}{\left(x \right)}}{3} + \csc{\left(x \right)}$ 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. Let $u = \csc{\left(x \right)}$.

      2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$

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

         1. The derivative of cosecant is negative cosecant times cotangent:

            $$\frac{d}{d x} \csc{\left(x \right)} = - \cot{\left(x \right)} \csc{\left(x \right)}$$

         The result of the chain rule is:

         $$- \frac{3 \cos{\left(x \right)} \csc^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

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

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

2. Now simplify:

   $$\frac{\cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$$

The answer is:

$$\frac{\cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$$