y=x+1⼀x在x=x0处的导数，定义法

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定义在x = a处的导数：f'(a) = lim(x-->a) [f'(x) - f'(a)]/(x - a)
y = x + 1/x
y'(x₀) = lim(x-->x₀) [(x + 1/x) - (x₀ + 1/x₀)]/(x - x₀)
= lim(x-->x₀) [(x² + 1)/x - (x₀² + 1)/x₀]/(x - x₀)
= lim(x-->x₀) [(x² + 1)x₀ - (x₀² + 1)x]/[xx₀(x - x₀)]
= lim(x-->x₀) (x₀x² + x₀ - xx₀² - x)/[xx₀(x - x₀)]
= lim(x-->x₀) [(x₀x - 1)x - (x₀x - 1)x₀]/[xx₀(x - x₀)]
= lim(x-->x₀) [(x₀x - 1)(x - x₀)]/[xx₀(x - x₀)]
= lim(x-->x₀) (x₀x - 1)/xx₀
= (x₀² - 1)/x₀²
= 1 - 1/x₀²

解：y的导数=△y/△x=(x+△x+1/(x+△x)-x-1/x)/△x=1-1/(x+△x)*x=1-1/(x0+0)x0=1-1/x0^2

y'=1-1/(x0^2)