Question

given the function $y = sqrt6{x}$, find $\frac{dy}{dx}$. express your answer in radical form without using negative exponents, simplifying all fractions. answer attempt 1 out of 2 $\frac{dy}{dx}=$

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt[6]{x}$ as $y=x^{\frac{1}{6}}$ using the rule $\sqrt[n]{a}=a^{\frac{1}{n}}$.

Step2: Apply the power - rule for differentiation

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

Step3: Simplify the exponent

Calculate $\frac{1}{6}-1=\frac{1 - 6}{6}=-\frac{5}{6}$. Then $\frac{dy}{dx}=\frac{1}{6}x^{-\frac{5}{6}}$.

Step4: Convert to radical form

Using the rule $a^{-n}=\frac{1}{a^{n}}$ and $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$, we get $\frac{dy}{dx}=\frac{1}{6x^{\frac{5}{6}}}=\frac{1}{6\sqrt[6]{x^{5}}}$.

Answer:

$\frac{1}{6\sqrt[6]{x^{5}}}$