Derivative of tan(t)

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:

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

3. Now simplify:

   $$\frac{1}{\cos^{2}{\left(t \right)}}$$

The answer is:

$$\frac{1}{\cos^{2}{\left(t \right)}}$$