Question

find the derivative of: $sqrt9{x}$
hint: recall that $sqrtn{x}=x^{\frac{1}{n}}$

Explanation:

Step1: Rewrite the function

We rewrite $\sqrt[9]{x}$ as $x^{\frac{1}{9}}$ using the hint $ \sqrt[n]{x}=x^{\frac{1}{n}}$.

Step2: Apply the power - rule for derivatives

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $y=x^{\frac{1}{9}}$, we have $n=\frac{1}{9}$. So $y^\prime=\frac{1}{9}x^{\frac{1}{9}-1}$.

Step3: Simplify the exponent

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

Answer:

$\frac{1}{9}x^{-\frac{8}{9}}$