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$f(x) = (x+16)/x$ and recall the difference quotient is \begin{align*} \frac{f(x+h) - f(x)}{h} \end{align*} Focusing on just the numerator for a second, we have \begin{align*} f(x+h)-f(x) &= \frac{x+h+16}{x+h}-\frac{x+16}{x} \\ &= \frac{x(x+h+16)}{x(x+h)}-\frac{(x+16)(x+h)}{x(x+h)} \\ &= \frac{x^2 + hx + 16x}{x(x+h)} - \frac{x^2 + 16x + hx + 16h}{x(x+h)} \\ &= \frac{-16h}{x(x+h)} \end{align*} Putting it all back together, \begin{align*} \frac{f(x+h) - f(x)}{h} &= \frac{\frac{-16h}{x(x+h)}}{h} \\ &= -\frac{16}{x(x+h)} \end{align*}
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Fri, 28 Sep 2018 15:54 GMT