※ 引述《verysong (verysong)》之銘言：
: f(x) = ｜ 1-x^2 ｜
:
: -2x(1-x^2)
: 證明 f'(x) = ____________
: ｜1-x^2｜
: 虛心請益
分開討論
(i) 0 ≦ |x| < 1
(-2x) * (1-x^2)
f(x) = 1 - x^2 => f'(x) = -2x = -------------------
1 - x^2
(-2x) * (1-x^2)
= -----------------
| 1 - x^2 |
(ii) |x| > 1
(-2x) * (1-x^2)
f(x) = x^2 - 1 => f'(x) = 2x = --------------------
x^2 - 1
(-2x) * (1-x^2)
= -----------------
| 1 - x^2 |
(iii) |x| = 1
f(1) = f(-1) = 0
f(x) - f(1) x^2 - 1
lim ---------------- = lim ------------ = 2
x → 1+ x - 1 x → 1+ x - 1
lim f(x) - f(1) 1 - x^2
x → 1- ------------- = lim ------------ = -2
x - 1 x → 1- x - 1
=> f 在 x = 1 不可微
同理 f 在 x = -1 不可微
(-2x) * (1-x^2)
∴ f'(x) = ------------------- , |x|≠1
| 1 - x^2 |
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 122.127.98.50