Derivative of x^5*cos(x)

1. Apply the product rule:

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

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

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

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

2. Now simplify:

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

The answer is: