Given: A function's derivative is given:
[tex]f^{\prime}(x)=\frac{3(3x-1)(x+1)}{2x^{\frac{3}{2}}}[/tex]
Required: To determine the value of f'(3).
Explanation: The value of f'(3) can be determined by substituting x=3 in f'(x) as follows-
[tex]f^{\prime}(3)=\frac{3[(3\times3)-1](3+1)}{2\times3^{\frac{3}{2}}}[/tex]
Further solving,
[tex]\begin{gathered} f^{\prime}(3)=\frac{3\times8\times4}{2\times3^{\frac{3}{2}}} \\ =\frac{16}{3^{\frac{1}{2}}} \end{gathered}[/tex]
On further solving the equation, we get
[tex]\begin{gathered} f^{\prime}(3)=\frac{16}{1.7321} \\ =9.2376 \end{gathered}[/tex]
Final Answer:
[tex]f^{\prime}(3)=9.2376[/tex]
