Derivative of y=(x-2)^2e^x-6

1. Differentiate $e^{x} \left(x - 2\right)^{2} - 6$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = \left(x - 2\right)^{2}$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Let $u = x - 2$.

      2. Apply the power rule: $u^{2}$ goes to $2 u$

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

         1. Differentiate $x - 2$ term by term:

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

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

            The result is: $1$

         The result of the chain rule is:

         $$2 x - 4$$

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

      1. The derivative of $e^{x}$ is itself.

      The result is: $\left(x - 2\right)^{2} e^{x} + \left(2 x - 4\right) e^{x}$

   2. The derivative of the constant $-6$ is zero.

   The result is: $\left(x - 2\right)^{2} e^{x} + \left(2 x - 4\right) e^{x}$

2. Now simplify:

   $$x \left(x - 2\right) e^{x}$$

The answer is: