Derivative of ctg(x/3)

1. There are multiple ways to do this derivative.

   Method #1

   1. Rewrite the function to be differentiated:

      $$\cot{\left(\frac{x}{3} \right)} = \frac{1}{\tan{\left(\frac{x}{3} \right)}}$$

   2. Let $u = \tan{\left(\frac{x}{3} \right)}$.

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

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

      1. Rewrite the function to be differentiated:

         $$\tan{\left(\frac{x}{3} \right)} = \frac{\sin{\left(\frac{x}{3} \right)}}{\cos{\left(\frac{x}{3} \right)}}$$

      2. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = \sin{\left(\frac{x}{3} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{3} \right)}$.

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

         1. Let $u = \frac{x}{3}$.

         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 $\frac{d}{d x} \frac{x}{3}$:

            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: $\frac{1}{3}$

            The result of the chain rule is:

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

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

         1. Let $u = \frac{x}{3}$.

         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 $\frac{d}{d x} \frac{x}{3}$:

            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: $\frac{1}{3}$

            The result of the chain rule is:

            $$- \frac{\sin{\left(\frac{x}{3} \right)}}{3}$$

         Now plug in to the quotient rule:

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

      The result of the chain rule is:

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

   Method #2

   1. Rewrite the function to be differentiated:

      $$\cot{\left(\frac{x}{3} \right)} = \frac{\cos{\left(\frac{x}{3} \right)}}{\sin{\left(\frac{x}{3} \right)}}$$

   2. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = \cos{\left(\frac{x}{3} \right)}$ and $g{\left(x \right)} = \sin{\left(\frac{x}{3} \right)}$.

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

      1. Let $u = \frac{x}{3}$.

      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 $\frac{d}{d x} \frac{x}{3}$:

         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: $\frac{1}{3}$

         The result of the chain rule is:

         $$- \frac{\sin{\left(\frac{x}{3} \right)}}{3}$$

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

      1. Let $u = \frac{x}{3}$.

      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 $\frac{d}{d x} \frac{x}{3}$:

         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: $\frac{1}{3}$

         The result of the chain rule is:

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

      Now plug in to the quotient rule:

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

2. Now simplify:

   $$- \frac{1}{3 \sin^{2}{\left(\frac{x}{3} \right)}}$$

The answer is:

$$- \frac{1}{3 \sin^{2}{\left(\frac{x}{3} \right)}}$$