Derivative of y=x^3sinx+3x^2cosx

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

   1. Apply the product rule:

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

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

      1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$

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

      1. The derivative of sine is cosine:

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

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

   2. Apply the product rule:

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

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

      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: $6 x$

      $g{\left(x \right)} = \cos{\left(x \right)}$; 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: $- 3 x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)}$

   The result is: $x^{3} \cos{\left(x \right)} + 6 x \cos{\left(x \right)}$

2. Now simplify:

   $$x \left(x^{2} + 6\right) \cos{\left(x \right)}$$

The answer is:

$$x \left(x^{2} + 6\right) \cos{\left(x \right)}$$