Derivative of log(sin(5*x))

1. Let $u = \sin{\left(5 x \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$:

   1. Let $u = 5 x$.

   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 $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$5 \cos{\left(5 x \right)}$$

   The result of the chain rule is:

   $$\frac{5 \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$$

4. Now simplify:

   $$\frac{5}{\tan{\left(5 x \right)}}$$

The answer is:

$$\frac{5}{\tan{\left(5 x \right)}}$$