Derivative of y=x^2∙sin(5x)+2

1. Differentiate $x^{2} \sin{\left(5 x \right)} + 2$ term by term:

   1. Apply the product rule:

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

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

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

      $g{\left(x \right)} = \sin{\left(5 x \right)}$; 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: $5 x^{2} \cos{\left(5 x \right)} + 2 x \sin{\left(5 x \right)}$

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

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

2. Now simplify:

   $$x \left(5 x \cos{\left(5 x \right)} + 2 \sin{\left(5 x \right)}\right)$$

The answer is:

$$x \left(5 x \cos{\left(5 x \right)} + 2 \sin{\left(5 x \right)}\right)$$