Derivative of (3x^2+2x-2)*sin(5x)

1. Apply the product rule:

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

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

   1. Differentiate $3 x^{2} + 2 x - 2$ 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: $x^{2}$ goes to $2 x$

         So, the result is: $6 x$

      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: $2$

      3. The derivative of the constant $\left(-1\right) 2$ is zero.

      The result is: $6 x + 2$

   $g{\left(x \right)} = \sin{\left(5 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 5 x$.

   2. The derivative of sine is cosine:

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$5 \cos{\left(5 x \right)}$$

   The result is: $\left(6 x + 2\right) \sin{\left(5 x \right)} + 5 \cdot \left(3 x^{2} + 2 x - 2\right) \cos{\left(5 x \right)}$

2. Now simplify:

   $$\left(6 x + 2\right) \sin{\left(5 x \right)} + \left(15 x^{2} + 10 x - 10\right) \cos{\left(5 x \right)}$$

The answer is:

$$\left(6 x + 2\right) \sin{\left(5 x \right)} + \left(15 x^{2} + 10 x - 10\right) \cos{\left(5 x \right)}$$