11. if ( f(x)=x^{\frac{3}{2}} ), then ( f^{prime}(4)= ) a -6 b -3 c 3 d 6 e 8
11. if ( f(x)=x^{\frac{3}{2}} ), then ( f^{prime}(4)= ) a -6 b -3 c 3 d 6 e 8
Answer
Explanation:
Step1: Differentiate the function
Use the power rule ( (x^n)^\prime=nx^{n - 1} ). For ( f(x)=x^{\frac{3}{2}} ), ( f^\prime(x)=\frac{3}{2}x^{\frac{3}{2}-1}=\frac{3}{2}x^{\frac{1}{2}} ).
Step2: Substitute ( x = 4 ) into the derivative
When ( x = 4 ), ( f^\prime(4)=\frac{3}{2}\times4^{\frac{1}{2}} ). Since ( 4^{\frac{1}{2}}=\sqrt{4} = 2 ), then ( f^\prime(4)=\frac{3}{2}\times2=3 ).
Answer:
C. 3