Question

given $g(x) = \frac{3}{x + 5}$, evaluate and simplify: $\frac{g(1 + h) - g(1)}{h} = $

Explanation:

Step1: Find \( g(1 + h) \)

Substitute \( x = 1 + h \) into \( g(x)=\frac{3}{x + 5} \), we get \( g(1 + h)=\frac{3}{(1 + h)+5}=\frac{3}{h + 6} \)

Step2: Find \( g(1) \)

Substitute \( x = 1 \) into \( g(x)=\frac{3}{x + 5} \), we get \( g(1)=\frac{3}{1 + 5}=\frac{3}{6}=\frac{1}{2} \)

Step3: Calculate \( g(1 + h)-g(1) \)

\( g(1 + h)-g(1)=\frac{3}{h + 6}-\frac{1}{2}=\frac{6-(h + 6)}{2(h + 6)}=\frac{6 - h - 6}{2(h + 6)}=\frac{-h}{2(h + 6)} \)

Step4: Divide by \( h \)

\( \frac{g(1 + h)-g(1)}{h}=\frac{\frac{-h}{2(h + 6)}}{h}=\frac{-h}{2(h + 6)}\times\frac{1}{h}=\frac{-1}{2(h + 6)} \) (assuming \( h
eq0 \))

Answer:

\( \frac{-1}{2(h + 6)} \)