Question

question given the function $y = \frac{5}{sqrt6{x}}$, find $\frac{dy}{dx}$. express your answer in radical form without using negative exponents, simplifying all fractions. answer attempt 1 out of 2 $\frac{dy}{dx}=$ submit answer watch video show examples

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{5}{\sqrt[6]{x}}$ as $y = 5x^{-\frac{1}{6}}$.

Step2: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $\frac{dy}{dx}=anx^{n - 1}$. Here, $a = 5$ and $n=-\frac{1}{6}$. So, $\frac{dy}{dx}=5\times(-\frac{1}{6})x^{-\frac{1}{6}-1}$.

Step3: Simplify the exponent

$-\frac{1}{6}-1=-\frac{1 + 6}{6}=-\frac{7}{6}$. So, $\frac{dy}{dx}=-\frac{5}{6}x^{-\frac{7}{6}}$.

Step4: Rewrite without negative exponents

$x^{-\frac{7}{6}}=\frac{1}{x^{\frac{7}{6}}}=\frac{1}{\sqrt[6]{x^{7}}}$. Thus, $\frac{dy}{dx}=-\frac{5}{6\sqrt[6]{x^{7}}}$.

Answer:

$-\frac{5}{6\sqrt[6]{x^{7}}}$