Derivative of ln(x+5)^9

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

2. Apply the power rule: $u^{9}$ goes to $9 u^{8}$

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

   1. Let $u = x + 5$.

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

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

      1. Differentiate $x + 5$ term by term:

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

         2. The derivative of the constant $5$ is zero.

         The result is: $1$

      The result of the chain rule is:

      $$\frac{1}{x + 5}$$

   The result of the chain rule is:

   $$\frac{9 \log{\left(x + 5 \right)}^{8}}{x + 5}$$

4. Now simplify:

   $$\frac{9 \log{\left(x + 5 \right)}^{8}}{x + 5}$$

The answer is: