Derivative of y=(x5+3x)*sinx

1. Apply the product rule:

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

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

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

      1. The derivative of the constant $x_{5}$ is zero.

      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: $3$

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

The answer is:

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