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

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{5} + 3 x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $x^{5} + 3 x$ term by term:

      1. Apply the power rule: $x^{5}$ goes to $5 x^{4}$

      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: $3$

      The result is: $5 x^{4} + 3$

   $g{\left(x \right)} = \sin{\left(x \right)}$; 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: $\left(5 x^{4} + 3\right) \sin{\left(x \right)} + \left(x^{5} + 3 x\right) \cos{\left(x \right)}$

2. Now simplify:

   $$x \left(x^{4} + 3\right) \cos{\left(x \right)} + \left(5 x^{4} + 3\right) \sin{\left(x \right)}$$

The answer is:

$$x \left(x^{4} + 3\right) \cos{\left(x \right)} + \left(5 x^{4} + 3\right) \sin{\left(x \right)}$$