Question

question 22
find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify it.
$f(x)=\frac{5}{x}$
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Explanation:

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

Given \(f(x)=\frac{5}{x}\), then \(f(x + h)=\frac{5}{x + h}\)

Step2: Substitute into difference - quotient formula

\(\frac{f(x + h)-f(x)}{h}=\frac{\frac{5}{x + h}-\frac{5}{x}}{h}\)

Step3: Get a common denominator for the numerator

\(\frac{\frac{5x-5(x + h)}{x(x + h)}}{h}=\frac{\frac{5x-5x-5h}{x(x + h)}}{h}\)

Step4: Simplify the numerator

\(\frac{\frac{- 5h}{x(x + h)}}{h}\)

Step5: Divide by \(h\)

\(\frac{-5h}{x(x + h)}\times\frac{1}{h}=-\frac{5}{x(x + h)}\)

Answer:

\(-\frac{5}{x(x + h)}\)