Question

question 10 (3 points)
use the first principles definition to determine the derivative of the general
reciprocal function $f(x)=\frac{1}{ax + b}$.

Explanation:

Step1: Recall the first - principles formula

The derivative of a function \(y = f(x)\) using the first - principles definition is \(f^{\prime}(x)=\lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}\). Given \(f(x)=\frac{1}{ax + b}\), then \(f(x + h)=\frac{1}{a(x + h)+b}=\frac{1}{ax+ah + b}\).

Step2: Substitute into the first - principles formula

$$
LATEXBLOCK0
$$

Step3: Simplify and find the limit

Cancel out the \(h\) terms (\(h
eq0\) when simplifying the fraction). Then \(\lim_{h
ightarrow0}\frac{-ah}{h(ax + b)(ax+ah + b)}=\lim_{h
ightarrow0}\frac{-a}{(ax + b)(ax+ah + b)}\).
As \(h
ightarrow0\), we substitute \(h = 0\) into the expression \(\frac{-a}{(ax + b)(ax+ah + b)}\).

Answer:

\(f^{\prime}(x)=-\frac{a}{(ax + b)^{2}}\)