Question

一道导数题

利用导数的定义，求出函数y=x+1/x的导数，并据此求函数在x=1处的导数

要具体过程

Answer 1

y=x+1/x
y'=lim[h→0] {[(x+h)+1/(x+h)]-(x+1/x)}/h
=lim[h→0] [(x+h)²+1]/(x+h)-(x²+1)/x]/h
=lim[h→0] [(x²+2xh+h²+1)/(x+h)-(x²+1)/x]/h
=lim[h→0] {[x(x²+2xh+h²+1)-(x+h)(x²+1)]/(x²+xh)}/h
=lim[h→0] [(x³+2x²h+xh²+x-x³-h-x²h-x)/(x²+xh)]/h
=lim[h→0] [(x²h+xh²-h)/(x²+xh)]/h
=lim[h→0] (x²+xh-1)/(x²+xh)
=(x²-1)/x²
=1-1/x²

在x=1处的导数y'(1)=1-1/1²=0