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 formula

$$
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$$

Step3: Simplify the expression

Cancel out the \(h\) terms: \(\frac{-ah}{h(ax + b)(ax+ah + b)}=\frac{-a}{(ax + b)(ax+ah + b)}\)

Step4: Take the limit as \(h

ightarrow0\)
\(\lim_{h
ightarrow0}\frac{-a}{(ax + b)(ax+ah + b)}\). As \(h
ightarrow0\), \(ax+ah + b
ightarrow ax + b\). So \(f^{\prime}(x)=\frac{-a}{(ax + b)^2}\)

Answer:

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