Derivative of y=(x-1)exp(ax)

1. Apply the product rule:

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

   $f{\left(x \right)} = x - 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $x - 1$ term by term:

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

      2. The derivative of the constant $-1$ is zero.

      The result is: $1$

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

   1. Let $u = a x$.

   2. The derivative of $e^{u}$ is itself.

   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:

      $$a e^{a x}$$

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

2. Now simplify:

   $$\left(a \left(x - 1\right) + 1\right) e^{a x}$$

The answer is:

$$\left(a \left(x - 1\right) + 1\right) e^{a x}$$