Derivative of tg(x\x)

1. Rewrite the function to be differentiated:

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

2. Apply the quotient rule, which is:

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

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

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

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

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

      1. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = x$ and $g{\left(x \right)} = x$.

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

         1. Apply the power rule: $x$ goes to $1$

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

         1. Apply the power rule: $x$ goes to $1$

         Now plug in to the quotient rule:

         $0$

      The result of the chain rule is:

      $$0$$

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

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

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

      1. Apply the quotient rule, which is:

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

         $f{\left(x \right)} = x$ and $g{\left(x \right)} = x$.

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

         1. Apply the power rule: $x$ goes to $1$

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

         1. Apply the power rule: $x$ goes to $1$

         Now plug in to the quotient rule:

         $0$

      The result of the chain rule is:

      $$0$$

   Now plug in to the quotient rule:

   $0$

The answer is:

$$0$$