Derivative of tg(x/2)

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:

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

3. Now simplify:

   $$\frac{1}{\cos{\left(x \right)} + 1}$$

The answer is:

$$\frac{1}{\cos{\left(x \right)} + 1}$$