Derivative of tan(3*y)

1. Rewrite the function to be differentiated:

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

2. Apply the quotient rule, which is:

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

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

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

   1. Let $u = 3 y$.

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

      1. The derivative of a constant times a function is the constant times the derivative of the function.

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

         So, the result is: $3$

      The result of the chain rule is:

      $$3 \cos{\left(3 y \right)}$$

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

   1. Let $u = 3 y$.

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

      1. The derivative of a constant times a function is the constant times the derivative of the function.

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

         So, the result is: $3$

      The result of the chain rule is:

      $$- 3 \sin{\left(3 y \right)}$$

   Now plug in to the quotient rule:

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

3. Now simplify:

   $$\frac{3}{\cos^{2}{\left(3 y \right)}}$$

The answer is:

$$\frac{3}{\cos^{2}{\left(3 y \right)}}$$