Derivative of (3-2*x)*cos(x)+2*sin(x)+4

1. Differentiate $\left(3 - 2 x\right) \cos{\left(x \right)} + 2 \sin{\left(x \right)} + 4$ term by term:

   1. Apply the product rule:

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

      $f{\left(x \right)} = 3 - 2 x$; 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. 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: $2$

            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. The derivative of a constant times a function is the constant times the derivative of the function.

      1. The derivative of sine is cosine:

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

      So, the result is: $2 \cos{\left(x \right)}$

   3. The derivative of the constant $4$ is zero.

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

2. Now simplify:

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

The answer is:

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