Question

find the derivative of the function (y = sqrt9{x^{8}}-x^{pi})

Explanation:

Step1: Rewrite the first - term

Rewrite $\sqrt[9]{x^{8}}$ as $x^{\frac{8}{9}}$. So $y = x^{\frac{8}{9}}-x^{\pi}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$.
For the first term $u=x^{\frac{8}{9}}$, its derivative $u^\prime=\frac{8}{9}x^{\frac{8}{9}-1}=\frac{8}{9}x^{-\frac{1}{9}}$.
For the second term $v = x^{\pi}$, its derivative $v^\prime=\pi x^{\pi - 1}$.

Step3: Find the derivative of y

Since $y = u - v$, then $y^\prime=u^\prime - v^\prime$.
$y^\prime=\frac{8}{9}x^{-\frac{1}{9}}-\pi x^{\pi - 1}=\frac{8}{9x^{\frac{1}{9}}}-\pi x^{\pi - 1}$.

Answer:

$y^\prime=\frac{8}{9x^{\frac{1}{9}}}-\pi x^{\pi - 1}$