question given the function y = -\\frac{5\\sqrt{x^{5}}}{3}, find \\frac{dy}{dx}. express your answer in…

question given the function y = -\\frac{5\\sqrt{x^{5}}}{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}=

question given the function y = -\\frac{5\\sqrt{x^{5}}}{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}=

Answer

Explanation:

Step1: Rewrite the function

First, rewrite $y =-\frac{5\sqrt{x^{5}}}{3}=-\frac{5x^{\frac{5}{2}}}{3}$.

Step2: Apply power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $\frac{dy}{dx}=anx^{n - 1}$. Here, $a=-\frac{5}{3}$ and $n=\frac{5}{2}$. So, $\frac{dy}{dx}=-\frac{5}{3}\times\frac{5}{2}x^{\frac{5}{2}-1}$.

Step3: Simplify the exponent and the coefficient

Calculate $\frac{5}{2}-1=\frac{5 - 2}{2}=\frac{3}{2}$, and $-\frac{5}{3}\times\frac{5}{2}=-\frac{25}{6}$. So, $\frac{dy}{dx}=-\frac{25}{6}x^{\frac{3}{2}}$.

Step4: Convert back to radical form

Since $x^{\frac{3}{2}}=\sqrt{x^{3}}$, then $\frac{dy}{dx}=-\frac{25\sqrt{x^{3}}}{6}$.

Answer:

$-\frac{25\sqrt{x^{3}}}{6}$