Derivative of -(ln(cos(3x)))/3

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

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

      1. Let $u = \cos{\left(3 x \right)}$.

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

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

         1. Let $u = 3 x$.

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

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

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

               So, the result is: $3$

            The result of the chain rule is:

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

         The result of the chain rule is:

         $$- \frac{3 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$$

      So, the result is: $\frac{3 \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$

   So, the result is: $\frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$

2. Now simplify:

   $$\tan{\left(3 x \right)}$$

The answer is:

$$\tan{\left(3 x \right)}$$