Derivative of y=ln(5x)/2x+3

1. Differentiate $\frac{x \log{\left(5 x \right)}}{2} + 3$ term by term:

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

      1. Apply the product rule:

         $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

         $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

         $g{\left(x \right)} = \log{\left(5 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

         1. Let $u = 5 x$.

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

         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:

            $$\frac{1}{x}$$

         The result is: $\log{\left(5 x \right)} + 1$

      So, the result is: $\frac{\log{\left(5 x \right)}}{2} + \frac{1}{2}$

   2. The derivative of the constant $3$ is zero.

   The result is: $\frac{\log{\left(5 x \right)}}{2} + \frac{1}{2}$

The answer is:

$$\frac{\log{\left(5 x \right)}}{2} + \frac{1}{2}$$