Derivative of (2x-3)cosx

1. Apply the product rule:

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

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

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

      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: $2$

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

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

2. Now simplify:

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

The answer is:

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