Derivative of tan(t)^(2)

1. Let $u = \tan{\left(t \right)}$.

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

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

   1. Rewrite the function to be differentiated:

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

   2. Apply the quotient rule, which is:

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

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

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

      1. The derivative of sine is cosine:

         $$\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}$$

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

      1. The derivative of cosine is negative sine:

         $$\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$$

      Now plug in to the quotient rule:

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

   The result of the chain rule is:

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

4. Now simplify:

   $$\frac{2 \tan{\left(t \right)}}{\cos^{2}{\left(t \right)}}$$

The answer is:

$$\frac{2 \tan{\left(t \right)}}{\cos^{2}{\left(t \right)}}$$