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

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

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

Answer

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)