例3 设f(x)= xx(x-2),求 f'(x)解先去掉绝对值x(x - 2),x≤0f(x) =^ -x(x- 2),0<x<2x(x-2),x≥2当x = 0时, f(0) = f(0) = 0, f'(0) = 0;当x >2或x< 0时,f'(x)= 3x2 -4x;当0 <x<2时, f(x)=-3x2 + 4x;
例3 解 先去掉绝对值 f (x) = ( 2), x 0 2 x x − ( 2), 0 x 2 2 − x x − ( 2), x 2 2 x x − 设f (x) = x x(x − 2),求 f (x). 当x = 0时, (0) = (0) = 0, − + f f f (0) = 0; 当0 x 2时, ( ) 3 4 ; 2 当x 2或x 0时, f x = x − x ( ) 3 4 ; 2 f x = − x + x
x(x-2),x≤0f(x) =3 - x(x -2),0 < x <2,当x = 2时,x(x-2),x ≥2x(x-2)f(x)- f(2)f'(2) = lim :lim-4x-2x-2x-→2x-→2x(x-2)f(x)- f(2)f'(2) = limlim:4.x-2x-2x-2x→2*，： f(x)在x =2处不可导f'(2) ± f’(2),[3x? - 4x,x>2,或x<0f'(x)=30, x=0,-3x2 +4x,, 0<x<2
当x = 2时, 2 ( ) (2) (2) lim 2 − − = − → − x f x f f x 2 ( 2) lim 2 2 − − − = → − x x x x = −4. 2 ( ) (2) (2) lim 2 − − = + → + x f x f f x 2 ( 2) lim 2 2 − − = → + x x x x = 4. (2) (2), − + f f = f x x ( ) 2 . 在 处不可导 x x x x f x x x x x 2 2 3 4 , 2, 0 ( ) 0, 0, 3 4 , 0 2, − = = − + 或 , ( 2), 2 ( 2),0 2 ( 2), 0 ( ) 2 2 2 − − − − = x x x x x x x x x f x