Question

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)