find the derivative of the function: $f(x)=\frac{1}{sqrt{x^{3}}}$

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=\frac{1}{\sqrt{x^{3}}} ) as ( f(x)=x^{-\frac{3}{2}} ) using the rule ( \frac{1}{x^{n}} = x^{-n} ) and ( \sqrt{x^{m}}=x^{\frac{m}{2}} ).

Step2: Apply the power rule

The power rule for differentiation is ( (x^{n})^\prime=nx^{n - 1} ). For ( y = x^{-\frac{3}{2}} ), then ( y^\prime=-\frac{3}{2}x^{-\frac{3}{2}-1}=-\frac{3}{2}x^{-\frac{5}{2}} ).

Step3: Rewrite in radical form

Using ( x^{-\frac{5}{2}}=\frac{1}{x^{\frac{5}{2}}}=\frac{1}{\sqrt{x^{5}}} ), we get ( y^\prime=-\frac{3}{2\sqrt{x^{5}}} ).

Answer:

( f^\prime(x)=-\frac{3}{2\sqrt{x^{5}}} ) (the fourth option)