Question

estimate the rate of change of (f(x)=\frac{3}{x}) at (x = 4)

Explanation:

Step1: Recall derivative formula

The derivative of $f(x)=\frac{a}{x}=ax^{-1}$ using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$ is $f^\prime(x)=-ax^{-2}=-\frac{a}{x^{2}}$. Here $a = 3$, so $f^\prime(x)=-\frac{3}{x^{2}}$.

Step2: Evaluate derivative at $x = 4$

Substitute $x = 4$ into $f^\prime(x)$. We have $f^\prime(4)=-\frac{3}{4^{2}}=-\frac{3}{16}$.

Answer:

$-\frac{3}{16}$