Derivative of log(sin(x/2))-cot(x/2)

1. Differentiate $\log{\left(\sin{\left(\frac{x}{2} \right)} \right)} - \cot{\left(\frac{x}{2} \right)}$ term by term:

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

   2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

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

      1. Let $u = \frac{x}{2}$.

      2. The derivative of sine is cosine:

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

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

         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: $\frac{1}{2}$

         The result of the chain rule is:

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

      The result of the chain rule is:

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

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

      1. There are multiple ways to do this derivative.

         Method #1

         1. Rewrite the function to be differentiated:

            $$\cot{\left(\frac{x}{2} \right)} = \frac{1}{\tan{\left(\frac{x}{2} \right)}}$$

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

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

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

            1. Rewrite the function to be differentiated:

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

            2. Apply the quotient rule, which is:

               $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

               $f{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$.

               To find $\frac{d}{d x} f{\left(x \right)}$:

               1. Let $u = \frac{x}{2}$.

               2. The derivative of sine is cosine:

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

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

                  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: $\frac{1}{2}$

                  The result of the chain rule is:

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

               To find $\frac{d}{d x} g{\left(x \right)}$:

               1. Let $u = \frac{x}{2}$.

               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 $\frac{d}{d x} \frac{x}{2}$:

                  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: $\frac{1}{2}$

                  The result of the chain rule is:

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

               Now plug in to the quotient rule:

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

            The result of the chain rule is:

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

         Method #2

         1. Rewrite the function to be differentiated:

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

         2. Apply the quotient rule, which is:

            $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

            $f{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$.

            To find $\frac{d}{d x} f{\left(x \right)}$:

            1. Let $u = \frac{x}{2}$.

            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 $\frac{d}{d x} \frac{x}{2}$:

               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: $\frac{1}{2}$

               The result of the chain rule is:

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

            To find $\frac{d}{d x} g{\left(x \right)}$:

            1. Let $u = \frac{x}{2}$.

            2. The derivative of sine is cosine:

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

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

               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: $\frac{1}{2}$

               The result of the chain rule is:

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

            Now plug in to the quotient rule:

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

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

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

2. Now simplify:

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

The answer is:

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