find the derivative of: $sqrt9{x}$\nhint: recall that $sqrtn{x}=x^{\frac{1}{n}}$
find the derivative of: $sqrt9{x}$\nhint: recall that $sqrtn{x}=x^{\frac{1}{n}}$
Answer
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:
$\frac{1}{9}x^{-\frac{8}{9}}$