2. find the derivative of $y = \\frac{x^{8}+1}{x}$. $\\frac{dy}{dx}=$

Explanation:

Step1: Simplify the function

First, rewrite $y=\frac{x^{8}+1}{x}$ as $y = x^{7}+\frac{1}{x}=x^{7}+x^{-1}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $\frac{dy}{dx}=nx^{n - 1}$. For $y=x^{7}$, $\frac{d}{dx}(x^{7})=7x^{6}$; for $y = x^{-1}$, $\frac{d}{dx}(x^{-1})=-1x^{-2}$.

Step3: Combine the derivatives

$\frac{dy}{dx}=\frac{d}{dx}(x^{7}+x^{-1})=\frac{d}{dx}(x^{7})+\frac{d}{dx}(x^{-1}) = 7x^{6}-x^{-2}=7x^{6}-\frac{1}{x^{2}}$.

Answer:

$7x^{6}-\frac{1}{x^{2}}$