Question

given that $f(x)=\frac{5}{x}$, write an expression in terms of $x$ and $h$ that represents the average rate of change of $f$ over any interval of length $h$. simplify your answer.
-5/(x(x + h))
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question 6. points possible: 4
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Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[x,x + h]$ is $\frac{f(x + h)-f(x)}{h}$.

Step2: Find $f(x + h)$

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

Step3: Substitute into the formula

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

Step4: Get a common denominator for the numerator

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

Step5: Simplify the numerator and the fraction

The numerator $5x-5x - 5h=-5h$. So, $\frac{-5h}{x(x + h)h}=-\frac{5}{x(x + h)}$.

Answer:

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