Derivative of y=(x-8)(sinx)

1. Apply the product rule:

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

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

   1. Differentiate $x - 8$ term by term:

      1. Apply the power rule: $x$ goes to $1$

      2. The derivative of the constant $-8$ is zero.

      The result is: $1$

   $g{\left(x \right)} = \sin{\left(x \right)}$; 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: $\left(x - 8\right) \cos{\left(x \right)} + \sin{\left(x \right)}$

2. Now simplify:

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

The answer is: