Derivative of y=ln(ax)+bx

1. Differentiate $b x + \log{\left(a x \right)}$ term by term:

   1. Let $u = a x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} a x$:

      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: $a$

      The result of the chain rule is:

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

   4. 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: $b$

   The result is: $b + \frac{1}{x}$

The answer is:

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