if ( f(x)=\frac{2}{x^{2}} ), find ( f^{prime}(4) ).
if ( f(x)=\frac{2}{x^{2}} ), find ( f^{prime}(4) ).
Answer
Explanation:
Step1: Rewrite the function
Rewrite ( f(x)=\frac{2}{x^{2}} ) as ( f(x) = 2x^{-2} ).
Step2: Apply the power rule
The power rule is ( (x^{n})^\prime=nx^{n - 1} ). For ( y = ax^{n} ), ( y^\prime=anx^{n - 1} ). Here ( a = 2 ) and ( n=-2 ), so ( f^\prime(x)=2\times(-2)x^{-2 - 1}=-4x^{-3}=-\frac{4}{x^{3}} ).
Step3: Substitute ( x = 4 )
Substitute ( x = 4 ) into ( f^\prime(x) ). ( f^\prime(4)=-\frac{4}{4^{3}}=-\frac{4}{64}=-\frac{1}{16} ).
Answer:
(-\frac{1}{16})