Question

find and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$, $h
eq0$ for the given function.
$f(x)=\frac{13}{x}$
$\frac{f(x + h)-f(x)}{h}=square$ (simplify your answer.)

Explanation:

Step1: Find \(f(x + h)\)

Substitute \(x+h\) into \(f(x)\): \(f(x + h)=\frac{13}{x + h}\)

Step2: Calculate \(f(x + h)-f(x)\)

\(f(x + h)-f(x)=\frac{13}{x + h}-\frac{13}{x}=\frac{13x-13(x + h)}{x(x + h)}=\frac{13x-13x-13h}{x(x + h)}=\frac{- 13h}{x(x + h)}\)

Step3: Divide by \(h\)

\(\frac{f(x + h)-f(x)}{h}=\frac{\frac{-13h}{x(x + h)}}{h}=\frac{-13h}{x(x + h)}\times\frac{1}{h}=-\frac{13}{x(x + h)}\)

Answer:

\(-\frac{13}{x(x + h)}\)