x的三分之一次方求导数 用导数的定义求解 求推导过程
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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以得到
(x+dx)^(1/3)
-x^(1/3)
=(x+dx
-x)
/
[(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以得到
(x+dx)^(1/3)
-x^(1/3)
=(x+dx
-x)
/
[(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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