Derivative of y=x^8(cosx-3)

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{8}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{8}$ goes to $8 x^{7}$

   $g{\left(x \right)} = \cos{\left(x \right)} - 3$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $\cos{\left(x \right)} - 3$ term by term:

      1. The derivative of cosine is negative sine:

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

      2. The derivative of the constant $\left(-1\right) 3$ is zero.

      The result is: $- \sin{\left(x \right)}$

   The result is: $- x^{8} \sin{\left(x \right)} + 8 x^{7} \left(\cos{\left(x \right)} - 3\right)$

2. Now simplify:

   $$x^{7} \left(- x \sin{\left(x \right)} + 8 \cos{\left(x \right)} - 24\right)$$

The answer is:

$$x^{7} \left(- x \sin{\left(x \right)} + 8 \cos{\left(x \right)} - 24\right)$$