Derivative of (3-x)exp^(x-2)

1. Apply the product rule:

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

   $f{\left(x \right)} = 3 - x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $3 - x$ term by term:

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

      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: $-1$

      The result is: $-1$

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

   1. Let $u = x - 2$.

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

   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:

      $$e^{x - 2}$$

   The result is: $\left(3 - x\right) e^{x - 2} - e^{x - 2}$

2. Now simplify:

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

The answer is:

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