Derivative of ln*tg^2(x/6)

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = \tan^{2}{\left(\frac{x}{6} \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \tan{\left(\frac{x}{6} \right)}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{x}{6} \right)}$:

      1. Rewrite the function to be differentiated:

         $$\tan{\left(\frac{x}{6} \right)} = \frac{\sin{\left(\frac{x}{6} \right)}}{\cos{\left(\frac{x}{6} \right)}}$$

      2. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = \sin{\left(\frac{x}{6} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{6} \right)}$.

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

         1. Let $u = \frac{x}{6}$.

         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 $\frac{d}{d x} \frac{x}{6}$:

            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: $\frac{1}{6}$

            The result of the chain rule is:

            $$\frac{\cos{\left(\frac{x}{6} \right)}}{6}$$

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

         1. Let $u = \frac{x}{6}$.

         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 $\frac{d}{d x} \frac{x}{6}$:

            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: $\frac{1}{6}$

            The result of the chain rule is:

            $$- \frac{\sin{\left(\frac{x}{6} \right)}}{6}$$

         Now plug in to the quotient rule:

         $\frac{\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}}{\cos^{2}{\left(\frac{x}{6} \right)}}$

      The result of the chain rule is:

      $$\frac{2 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right) \tan{\left(\frac{x}{6} \right)}}{\cos^{2}{\left(\frac{x}{6} \right)}}$$

   The result is: $\frac{2 \left(\frac{\sin^{2}{\left(\frac{x}{6} \right)}}{6} + \frac{\cos^{2}{\left(\frac{x}{6} \right)}}{6}\right) \log{\left(x \right)} \tan{\left(\frac{x}{6} \right)}}{\cos^{2}{\left(\frac{x}{6} \right)}} + \frac{\tan^{2}{\left(\frac{x}{6} \right)}}{x}$

2. Now simplify:

   $$\frac{\left(x \log{\left(x \right)} + \frac{3 \sin{\left(\frac{x}{3} \right)}}{2}\right) \tan{\left(\frac{x}{6} \right)}}{3 x \cos^{2}{\left(\frac{x}{6} \right)}}$$

The answer is: