Question

if ( f(x)=\frac{2}{x^{2}} ), find ( f^{prime}(4) ).

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:

\(-\frac{1}{16}\)