Question

use the power rule to find the derivative of f(x) = $sqrt6{x}$.
f(x) = $\frac{1}{6sqrt6{x^{5}}}$

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\sqrt[6]{x}$ as $f(x)=x^{\frac{1}{6}}$.

Step2: Apply the power - rule

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $f(x)=x^{\frac{1}{6}}$, we have $n=\frac{1}{6}$. So $f^\prime(x)=\frac{1}{6}x^{\frac{1}{6}-1}$.

Step3: Simplify the exponent

$\frac{1}{6}-1=\frac{1 - 6}{6}=-\frac{5}{6}$. So $f^\prime(x)=\frac{1}{6}x^{-\frac{5}{6}}$.

Step4: Rewrite in radical form

Since $x^{-\frac{5}{6}}=\frac{1}{x^{\frac{5}{6}}}=\frac{1}{\sqrt[6]{x^{5}}}$, then $f^\prime(x)=\frac{1}{6\sqrt[6]{x^{5}}}$.

Answer:

$f^\prime(x)=\frac{1}{6\sqrt[6]{x^{5}}}$