隱函數的微分
(甲) 隱函數的微分
2x^2+xy+y^2- 4 = 0 對其做x的微分, 可得
4x + y + xy' + 2yy' = 0
(x+2y)y' = - (4x+y)
y' = -(4x+y) / (x+2y)
dy/dx| (x,y) = (-1,2) = -(-4+2)/ -1+4 = 2/3
所以其切線為 y- 2 = 2/3(x-(-1))
dy/dx 為對式中x做一次微分, 求得
xy'+y+2yy'-2x = 0
(x+2y)y' = 2x -y
y' = (2x - y) / (x+2y)
x的2次微分,求得
(1+2y')y' + (x+2y)y" = 2 - y'
(x+2y)y" = 2 - (2+2y)y'
y" = 2 [ 1- (1+y)y' ] / (x+2y) 再將 y' 代入
法2 :
將 y' 再對x微分一次, 使用 (f/g)' 也可以求得.
原式對x做一次微分, 可得:
3y^2y' + 3y + 3xy' + 3x^2 = 0
3(x+y^2)y' = -3(y+x^2)
(x+y^2)y' = - (y+x^2)
y ' = - (y+x^2) / (x+y^2)
原式對x做一次微分, 可得:
6x - 2(y+xy') - 2yy' = 0
6x - 2y - 2xy' - 2yy' = 0
6x - 2y - 2(x+y)y' = 0
6 - 0 - 2(1+0)y' = 0
-2y' = -6
y' = 3
(1) 2x - 2y - 2xy' - 6yy' + 2 - y' = 0
(2) 2 + 2 - 2y' +6y' + 2 - y' = 0 , 3y' = -6 , y' = -2
先求對x一次的微分.
2/3x^-1/3 + 2/3y^-1/3y' = 0
y' = - 2/3x^-1/3 / 2/3y^-1/3 = x^-1/3 / y^-1/3 = (y/x)^1/3 , m = 1 at x , y = (2^1/2)/4 .
y = mx + b , (2^1/2)/ 4 = (2^1/2)/ 4 + b , b = 0 , y = x .
先求x的微分,
2ax+b(y+xy') + 2yy' + d + ey' = 0
2ax + d + by + (bx+2y+e)y' = 0
(bx+2y+e)y' = - (2ax + d + by)
y' = - (2ax + d + by) / (bx+2y+e)
Again ?
To find the tangent line , we must find its slope at (-1,2) . the slope of curve is m .
4x + y+ xy' + 2yy' = 0
-(4x+y) = (x+2y)y'
y' = - (4x+y) /(x+2y)
m = - (-4+2) / (-1+4) = 2 / 3 , So the tangent line of it is y = 2/3 x + b
2 = -2/3 + b , b = 8/3 .
so y = 2/3x + 8/3 is the tangent line .
由題意, x- 4y + 11 = 0 為其切線. 假設其切點為(a,b)
-4y = - (x+11) , 4y = (x+11) , y = 1/4x + 11/4 . 所以 m = 1/4 at (a,b) .
接著求x的微分, 2x + 8yy' + 2 = 0 , 8yy' = -2x -2 , y' = -(x+1)/(4y) .
-(a+1)/4b = 1/4 , 4b = -4(a+1), b = -(a+1) . 將其帶入曲線式中
a^2 + 4(a+1)^2 + 2a - 19 = 0
a^2 + 4 (a^2+ 2a + 1) + 2a - 19 = 0
5a^2 + 10 a - 15 = 0
a^2 + 2a -3 = 0
(a+3)(a-1) = 0
a= - 3 or 1
a = -3 , b = 2 => 2 = -3/4 + 11/4
a = 1 , b = -2 => -2 = 1/4 + 11/4 不合
所以 (-3,2) 為所求.
先求對x一次的微分.
2x - 8yy' - 16y' = 0
-8(y+2)y' = -2x
y' = x/4(y+2)
令 y = mx +b , m為其切線 at (1,-2)'s slope .
- 2 = m + b
- (2+b) = m ,
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