Derivative of (x-3)cos(x)

1. Apply the product rule:

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

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

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

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

      2. The derivative of the constant $-3$ is zero.

      The result is: $1$

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

   1. The derivative of cosine is negative sine:

      $$\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$$

   The result is: $- \left(x - 3\right) \sin{\left(x \right)} + \cos{\left(x \right)}$

2. Now simplify:

   $$\left(3 - x\right) \sin{\left(x \right)} + \cos{\left(x \right)}$$

The answer is:

$$\left(3 - x\right) \sin{\left(x \right)} + \cos{\left(x \right)}$$