导数的定义是什么?y=1/(1+x)的导数怎么求
1个回答
合并这两句,就是你想用导数的定义求这个函数吧~
导数定义f'(x) = lim(h->0) [f(x + h) - f(x)]/h
f'(a) = lim(x->a) [f(x) - f(a)]/(x - a),就是函数在x = a处的导数,也即曲线在该点的斜率.
y = 1/(1 + x)
y' = lim(h->0) [f(x + h) - f(x)]/h
= lim(h->0) [1/(1 + x + h) - 1/(1 + x)]/h
= lim(h->0) [(1 + x) - (1 + x + h)]/[h(1 + x)(1 + x + h)]
= lim(h->0) - h/[h(1 + x)(1 + x + h)]
= lim(h->0) - 1/[(1 + x)(1 + x + h)]
= - 1/[(1 + x)(1 + x + 0)]
= - 1/(1 + x)²
其实用导数公式也可以.[(f(x))^n]' = n · [(f(x))^(n - 1)] · f'(x)
(1/(1 + x))' = ((1 + x)^(- 1))'
= (- 1)(1 + x)^(- 1 - 1) · (1 + x)'
= - (1 + x)^(- 2) · (1)
= - 1/(1 + x)²
