Question

given the function $f(x)=\frac{3}{sqrt{x^{3}}}$, find $f(x)$. express your answer in radical form without using negative exponents, simplifying all fractions.

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{3}{\sqrt{x^{3}}}$ as $f(x) = 3x^{-\frac{3}{2}}$.

Step2: Apply the power - rule

The power - rule for differentiation is $(x^n)'=nx^{n - 1}$. For $y = 3x^{-\frac{3}{2}}$, we have $f'(x)=3\times(-\frac{3}{2})x^{-\frac{3}{2}-1}$.

Step3: Simplify the exponent

$f'(x)=-\frac{9}{2}x^{-\frac{5}{2}}$.

Step4: Rewrite without negative exponents

$f'(x)=-\frac{9}{2x^{\frac{5}{2}}}=-\frac{9}{2\sqrt{x^{5}}}$.

Answer:

$-\frac{9}{2\sqrt{x^{5}}}$