Derivative of logtan(x²)

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

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

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

   1. Rewrite the function to be differentiated:

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

   2. Apply the quotient rule, which is:

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

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

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

      1. Let $u = x^{2}$.

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

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

         The result of the chain rule is:

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

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

      1. Let $u = x^{2}$.

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

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

         The result of the chain rule is:

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

      Now plug in to the quotient rule:

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

   The result of the chain rule is:

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

4. Now simplify:

   $$\frac{4 x}{\sin{\left(2 x^{2} \right)}}$$

The answer is: