1樓：匿名使用者
這個吧,要用因式分解的平方差,立方和和立方差公式,挺麻煩的
平方差:x�0�5-h�0�5=(x+h)(x-h)
立方和:x�0�6+h�0�6=(x+h)(x�0�5-xh+h�0�5)而(x+h)=(x^1/3+h^1/3)(x^2/3-x^1/3*h^1/3+h^2/3)
立方差:x�0�6-h�0�6=(x-h)(x�0�5+xh+h�0�5)而(x-h)=(x^1/3-h^1/3)(x^2/3+x^1/3*h^1/3+h^2/3)
過程中是乘以(x^2/3-x^1/3*h^1/3+h^2/3)和(x^1/3-h^1/3)(x^2/3+x^1/3*h^1/3+h^2/3)湊出(x+h)和(x-h)
過程繁複,這樣好看點吧:
y=x^(2/3),根據導數基本定義,f'(x)=lim(h→0) [f(x+h)-f(x)]/h
導數y'=lim(h→0) 1/h*[(x+h)^(2/3)-x^(2/3)]
=lim(h→0) 1/h*
=lim(h→0)
分子:[(x+h)^(1/3)+x^(1/3)]*[(x+h)^(1/3)-x^(1/3)],這裡是平方差公式
分母:h
=lim(h→0)
分子:[(x+h)^(1/3)+x^(1/3)][(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)],這裡是立方和公式
*[(x+h)^(1/3)-x^(1/3)][(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)],這裡是立方差公式
分母:h*[(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)]
=lim(h→0)
分子:[(x+h)+x)][(x+h)-x]
分母:h*[(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)]
=lim(h→0)
分子:2x+h
分母:[(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)],約去h
=1/[x^(2/3)+x^(2/3)+x^(2/3)][x^(2/3)-x^(2/3)+x^(2/3)]*(2x)
=1/[3x^(2/3)*x^(2/3)]*2x
=2/3*x/x^(4/3)
=2/3*1/x^(1/3)
=2/[3x^(1/3)],即導數為(三乘以x的立方根)分之(二)
若用導數公式(x^n)'=nx^(n-1)過程簡單多了,就是[x^(2/3)]'=(2/3)*x^(2/3-1)=(2/3)*x^(-1/3)=2/[3x^(1/3)]
2樓：匿名使用者
這個啊 ，，，，不是很會啊
用定義法求三次根號下x平方的導數
3樓：匿名使用者
y=x^(2/3),根據導數基本定義,f'(x)=lim(h→0) [f(x+h)-f(x)]/h
導數y'=lim(h→0) [(x+h)^(2/3)-x^(2/3)]/h
分子:[(x+h)^(1/3)+x^(1/3)]*[(x+h)^(1/3)-x^(1/3)],這裡是平方差公式
分母:h
=lim(h→0)
分子:[(x+h)^(1/3)+x^(1/3)][(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(1/3)-x^(1/3)][(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)],
分母:h*[(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)]
=lim(h→0)
分子:[(x+h)+x)][(x+h)-x]
分母:h*[(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)]
=lim(h→0)
分子:2x+h
分母:[(x+h)^(2/3)+(x+h)^(1/3)*x^(1/3)+x^(2/3)]*[(x+h)^(2/3)-(x+h)^(1/3)*x^(1/3)+x^(2/3)],約去h
=1/[x^(2/3)+x^(2/3)+x^(2/3)][x^(2/3)-x^(2/3)+x^(2/3)]*(2x)
=1/[3x^(2/3)*x^(2/3)]*2x
=2/3*x/x^(4/3)
=2/3*1/x^(1/3)
=2/[3x^(1/3)]
因此y=x^(2/3)的導數為2/3*x^(-1/3)
3次根號下x²的求導。
4樓：匿名使用者
解：(x^2/3)'=(2/3)x^(1-2/3)=(2/3)x^(-1/3).
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