Question

given the function (y = \frac{2}{sqrt4{x}}), find (\frac{dy}{dx}). express your answer in radical form without using negative exponents, simplifying all fractions.

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{2}{\sqrt[4]{x}}$ as $y = 2x^{-\frac{1}{4}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^n$, then $\frac{dy}{dx}=anx^{n - 1}$. Here, $a = 2$ and $n=-\frac{1}{4}$. So, $\frac{dy}{dx}=2\times(-\frac{1}{4})x^{-\frac{1}{4}-1}$.

Step3: Simplify the exponent and coefficient

First, simplify the coefficient: $2\times(-\frac{1}{4})=-\frac{1}{2}$. Then, simplify the exponent: $-\frac{1}{4}-1=-\frac{1 + 4}{4}=-\frac{5}{4}$. So, $\frac{dy}{dx}=-\frac{1}{2}x^{-\frac{5}{4}}$.

Step4: Rewrite without negative exponents and in radical form

$x^{-\frac{5}{4}}=\frac{1}{x^{\frac{5}{4}}}=\frac{1}{\sqrt[4]{x^{5}}}$. Then $\frac{dy}{dx}=-\frac{1}{2\sqrt[4]{x^{5}}}$.

Answer:

$-\frac{1}{2\sqrt[4]{x^{5}}}$