利用导数的定义求函数的导数

Question

f（x）=三次根号下x

Answer 1

f(x)=x^(1/3)
f'(x)=lim [f(x+△x)-f(x)]/△x=lim [(x+△x)^(1/3)-x^(1/3)]/△x
△x->0 △x->0
=lim [(x+△x)^(1/3)-x^(1/3)]*[(x+△x)^(2/3)+(x+△x)^(1/3)*x^(1/3)+x^(2/3)]/{[(x+△x)^(2/3)+(x+△x)^(1/3)*x^(1/3)+x^(2/3)]*△x}
△x->0
=lim (x+△x-x)/{[(x+△x)^(2/3)+(x+△x)^(1/3)*x^(1/3)+x^(2/3)]*△x}
△x->0
=lim 1/[(x+△x)^(2/3)+(x+△x)^(1/3)*x^(1/3)+x^(2/3)]
△x->0
=1/3*x^(-2/3)

Answer 2

函数y=1/√x的导数=(当△x->0)lim（1/√（x+△x）-1/√x）/△x
=(当△x->0)lim（√x-√（x+△x））/(△x√（x+△x）√x)
=(当△x->0)lim{-1/[√（x+△x）√x(√x+√（x+△x))]}
=-1/(x(√x+√x))
=-1/(2x^(3/2))
=-x^(-3/2)/2