Derivative of y=log5(3x+2)

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

   1. Let $u = 3 x + 2$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 2\right)$:

      1. Differentiate $3 x + 2$ 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 $2$ is zero.

         The result is: $3$

      The result of the chain rule is:

      $$\frac{3}{3 x + 2}$$

   So, the result is: $\frac{3}{\left(3 x + 2\right) \log{\left(5 \right)}}$

2. Now simplify:

   $$\frac{3}{\left(3 x + 2\right) \log{\left(5 \right)}}$$

The answer is:

$$\frac{3}{\left(3 x + 2\right) \log{\left(5 \right)}}$$