Question

given the function y = 4/(5√(x³)), find dy/dx. express your answer in radical form without using negative exponents, simplifying all fractions.

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{4}{5\sqrt{x^{3}}}$ as $y=\frac{4}{5}x^{-\frac{3}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = ax^{n}$, then $\frac{dy}{dx}=anx^{n - 1}$. Here, $a=\frac{4}{5}$ and $n=-\frac{3}{2}$. So, $\frac{dy}{dx}=\frac{4}{5}\times(-\frac{3}{2})x^{-\frac{3}{2}-1}$.

Step3: Simplify the exponent and coefficient

First, simplify the coefficient: $\frac{4}{5}\times(-\frac{3}{2})=-\frac{12}{10}=-\frac{6}{5}$. Then, simplify the exponent: $-\frac{3}{2}-1=-\frac{3 + 2}{2}=-\frac{5}{2}$. So, $\frac{dy}{dx}=-\frac{6}{5}x^{-\frac{5}{2}}$.

Step4: Rewrite without negative exponents

$x^{-\frac{5}{2}}=\frac{1}{x^{\frac{5}{2}}}=\frac{1}{\sqrt{x^{5}}}$. So, $\frac{dy}{dx}=-\frac{6}{5\sqrt{x^{5}}}$.

Answer:

$-\frac{6}{5\sqrt{x^{5}}}$