Question

question given the function $y =-\frac{sqrt5{x^{4}}}{3}$, 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 $y =-\frac{3}{\sqrt[5]{x^{4}}}$ as $y=-3x^{-\frac{4}{5}}$ using the rule $\frac{1}{x^{n}}=x^{-n}$ and $\sqrt[m]{x^{n}} = x^{\frac{n}{m}}$.

Step2: Apply power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $\frac{dy}{dx}=anx^{n - 1}$. Here, $a=-3$ and $n =-\frac{4}{5}$. So, $\frac{dy}{dx}=-3\times(-\frac{4}{5})x^{-\frac{4}{5}-1}$.

Step3: Simplify the exponent and the coefficient

First, calculate the coefficient: $-3\times(-\frac{4}{5})=\frac{12}{5}$. Then, simplify the exponent: $-\frac{4}{5}-1=-\frac{4 + 5}{5}=-\frac{9}{5}$. So, $\frac{dy}{dx}=\frac{12}{5}x^{-\frac{9}{5}}$.

Step4: Rewrite without negative exponents

Using the rule $x^{-n}=\frac{1}{x^{n}}$, we get $\frac{dy}{dx}=\frac{12}{5x^{\frac{9}{5}}}$. And then rewrite in radical form: $\frac{dy}{dx}=\frac{12}{5\sqrt[5]{x^{9}}}$.

Answer:

$\frac{12}{5\sqrt[5]{x^{9}}}$