find and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$, $h\neq0$ for the given…

Explanation:

Step1: Find f(x + h)

Given (f(x)=\frac{14}{x}), then (f(x + h)=\frac{14}{x + h})

Step2: Substitute f(x + h) and f(x) into the difference - quotient formula

(\frac{f(x + h)-f(x)}{h}=\frac{\frac{14}{x + h}-\frac{14}{x}}{h})

Step3: Get a common denominator for the numerator

(\frac{14}{x + h}-\frac{14}{x}=\frac{14x-14(x + h)}{x(x + h)}=\frac{14x-14x-14h}{x(x + h)}=\frac{- 14h}{x(x + h)})

Step4: Simplify the difference - quotient

(\frac{\frac{-14h}{x(x + h)}}{h}=\frac{-14h}{x(x + h)}\times\frac{1}{h}=-\frac{14}{x(x + h)})

Answer:

(-\frac{14}{x(x + h)})