Question

用导数的定义计算y=1/x

Answer 1

设y=f(x)

1

f(x+△)=1/(x+△)

f(x)=1/x

f(x+△)-f(x)=1/(x+△)-1/x=(x-x-△)/x(x+△)=-△/x(x+△)

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

2

f(x+△)=(x+△+2)^3

f(x)=(x+2)^3

f(x+△)-f(x)=(x+△+2)^3-(x+2)^3

=[x+△+2-x-2][(x+△+2)^2+(x+△+2)(x+2)+(x+2)^2]

=△[(x+△+2)^2+(x+△+2)(x+2)+(x+2)^2]

f'(x)=lim△->0[f(x+△)-f(x)]/(x+△+2-x-2)

=lim△->0 △[(x+△+2)^2+(x+△+2)(x+2)+(x+2)^2] /△

=lim△->0 [(x+△+2)^2+(x+△+2)(x+2)+(x+2)^2]

=(x+2)^2+(x+2)^2+(x+2)^2

=3(x+2)^2