Question

find the derivative of the function: $f(x)=\sqrt{x^{5}}$ $f(x)=\frac{5}{2\sqrt{x^{3}}}$ $f(x)=-\frac{5\sqrt{x^{3}}}{2}$ $f(x)=\frac{5\sqrt{x^{3}}}{2}$ $f(x)=-\frac{5}{2\sqrt{x^{3}}}$

Explanation:

Step1: Simplify the function

Rewrite \( f(x)=\sqrt{x^{5}} \) as \( f(x)=x^{\frac{5}{2}} \) using the rule \( \sqrt[n]{a^{m}}=a^{\frac{m}{n}} \).

Step2: Apply the power rule

The power rule for differentiation is \( \frac{d}{dx}(x^{n}) = nx^{n - 1} \).
For \( y = x^{\frac{5}{2}} \), \( y^\prime=\frac{5}{2}x^{\frac{5}{2}-1}=\frac{5}{2}x^{\frac{3}{2}} \).
Rewrite \( x^{\frac{3}{2}}=\sqrt{x^{3}} \), so \( f^\prime(x)=\frac{5\sqrt{x^{3}}}{2} \).

Answer:

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