Derivative of (2sinx+3cosx)*x^2

1. Apply the product rule:

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

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

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

      1. 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: $2 \cos{\left(x \right)}$

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

         1. The derivative of cosine is negative sine:

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

         So, the result is: $- 3 \sin{\left(x \right)}$

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

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

   1. Apply the power rule: $x^{2}$ goes to $2 x$

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

2. Now simplify:

   $$x \left(x \left(- 3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 4 \sin{\left(x \right)} + 6 \cos{\left(x \right)}\right)$$

The answer is:

$$x \left(x \left(- 3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 4 \sin{\left(x \right)} + 6 \cos{\left(x \right)}\right)$$