Chapter 2 - The Derivative - 2.2 The Derivative Function - Exercises Set 2.2 - Page 131: 15

$\frac{dy}{dx} = -\frac{1}{x^2} $

$y = \frac{1}{x} $ $\frac{dy}{dx} = \lim\limits_{ Δx\to 0} \frac{f(x+Δx) - f(x)}{Δx} $ $\frac{dy}{dx} = \lim\limits_{ Δx\to 0} \frac{\frac{1}{x+Δx}-\frac{1}{x}}{Δx} $ $\frac{dy}{dx} = \lim\limits_{ Δx\to 0} \frac{1}{Δx}[\frac{1}{x+Δx} - \frac{1}{x}] $ $\frac{dy}{dx} = \lim\limits_{ Δx\to 0} \frac{1}{∆x} [\frac{x-(x+∆x)}{x(x+∆x)}] $ $\frac{dy}{dx} = \lim\limits_{ Δx\to 0} \frac{1}{∆x} [\frac{-∆x}{x(x+∆x)}] $ $\frac{dy}{dx} = \lim\limits_{ Δx\to 0} [\frac{-1}{x(x+∆x)}] $ $\frac{dy}{dx} = \frac{-1}{x(x+0)} $ $\frac{dy}{dx} = \frac{-1}{x^2} $ $\frac{dy}{dx} = -\frac{1}{x^2} $