14. if $f(x)=\frac{3}{x^{2}}$, find $f(4)$.
14. if $f(x)=\frac{3}{x^{2}}$, find $f(4)$.
Answer
Explanation:
Step1: Rewrite the function
Rewrite $f(x)=\frac{3}{x^{2}}$ as $f(x)=3x^{- 2}$.
Step2: Apply the power - rule for differentiation
The power - rule states that if $y = ax^{n}$, then $y'=anx^{n - 1}$. For $f(x)=3x^{-2}$, we have $f'(x)=3\times(-2)x^{-2 - 1}=-6x^{-3}=-\frac{6}{x^{3}}$.
Step3: Evaluate $f'(4)$
Substitute $x = 4$ into $f'(x)$. So $f'(4)=-\frac{6}{4^{3}}=-\frac{6}{64}=-\frac{3}{32}$.
Answer:
$-\frac{3}{32}$