用导数定义求y=x^（1/3）的导数
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y=x^(1/3)
那么
y'=lim(dx->0) [(x+dx)^(1/3) -x^(1/3)] /dx
注意由立方差公式可卜衫盯以得到型和
(x+dx)^(1/3) -x^(1/3)
=(x+dx -x) /塌搜 [(x+dx)^(2/3) + (x+dx)^(1/3)*x^(1/3) +x^(2/3)]
=dx / [(x+dx)^(2/3) + (x+dx)^(1/3)*x^(1/3) +x^(2/3)]
所以
y'=lim(dx->0) 1 / [(x+dx)^(2/3) + (x+dx)^(1/3)*x^(1/3) +x^(2/3)]
代入dx=0,
得到
y'= 1 /[x^(2/3) +x^(1/3)*x^(1/3) +x^(2/3)]
=1/3 *x^(-2/3)
那么
y'=lim(dx->0) [(x+dx)^(1/3) -x^(1/3)] /dx
注意由立方差公式可卜衫盯以得到型和
(x+dx)^(1/3) -x^(1/3)
=(x+dx -x) /塌搜 [(x+dx)^(2/3) + (x+dx)^(1/3)*x^(1/3) +x^(2/3)]
=dx / [(x+dx)^(2/3) + (x+dx)^(1/3)*x^(1/3) +x^(2/3)]
所以
y'=lim(dx->0) 1 / [(x+dx)^(2/3) + (x+dx)^(1/3)*x^(1/3) +x^(2/3)]
代入dx=0,
得到
y'= 1 /[x^(2/3) +x^(1/3)*x^(1/3) +x^(2/3)]
=1/3 *x^(-2/3)
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