Derivative of (3-2*x)*cos(x)

1. Apply the product rule:

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

   $f{\left(x \right)} = 3 - 2 x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $3 - 2 x$ 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: $-2$

      The result is: $-2$

   $g{\left(x \right)} = \cos{\left(x \right)}$; 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: $- \left(3 - 2 x\right) \sin{\left(x \right)} - 2 \cos{\left(x \right)}$

2. Now simplify:

   $$\left(2 x - 3\right) \sin{\left(x \right)} - 2 \cos{\left(x \right)}$$

The answer is:

$$\left(2 x - 3\right) \sin{\left(x \right)} - 2 \cos{\left(x \right)}$$