Derivative of y=xln(ax)-3x

1. Differentiate $x \log{\left(a x \right)} - 3 x$ term by term:

   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 $1$

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

      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:

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

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

   2. 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: $-3$

   The result is: $\log{\left(a x \right)} - 2$

The answer is:

$$\log{\left(a x \right)} - 2$$