Derivative of ln(3x+5)^2

1. Let $u = \log{\left(3 x + 5 \right)}$.

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

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

   1. Let $u = 3 x + 5$.

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

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

      1. Differentiate $3 x + 5$ term by term:

         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: $3$

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

         The result is: $3$

      The result of the chain rule is:

      $$\frac{3}{3 x + 5}$$

   The result of the chain rule is:

   $$\frac{6 \log{\left(3 x + 5 \right)}}{3 x + 5}$$

4. Now simplify:

   $$\frac{6 \log{\left(3 x + 5 \right)}}{3 x + 5}$$

The answer is: