推 ttst0720 : 謝謝您的分享 05/25 15:12
※ 引述《ttst0720 (心情陽光)》之銘言：
: 請問螢光邊框起來的部分是怎麼求出來的呢？
: https://i.imgur.com/9HoWLZi.jpg
1.
Tip: d(xy)= xdy + ydx
d(x^n) = nx^(n-1) dx
let u = y^(-1) => du = -y^(-2) dy
x -xdy dx -xdy + ydx
d(──) = d (xu) = xdu + udx = ──── + ─── = ──────
y y^2 y y^2
2.
1
Tip: d[ln(x)] = ─── dx
x
x
let u = ──
y
x -1 -y x
-d[ln(──)] = -d[ln(u)] = ──── du = ─── d(───)
y u x y
xdy -ydx
= ─────
xy
3.
y y
let u = ── , v= arctan(───) = arctan (u)
x x
=> u= tan(v) => du = sec^(2) (v)dv = (u^2 +1) dv
y 1
=> dv = d[arctan( ──) = ──── du
x u^2 +1
1 xdy-ydx xdy-ydx
= ───── X ──── = ──────
y^2 x^2 y^2 + x^2
── + 1
x^2
--
Logic can be patient for it is eternal. ----- Oliver Heaviside
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※ 編輯: Heaviside (220.128.239.58), 05/24/2019 23:17:27