Derivative of y=x^5sinx

1. Apply the product rule:

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

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

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

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

2. Now simplify:

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

The answer is: