Çözüldü Kapalı Fonksiyon Teoremi (Implicit Function Theorem) - Kısmi Türev (Partial Derivative)

a < 0, F(x, y) = x^3 + x·(y^2) - 4xy + 2y - 4 = 0 ⇒ F(1, a) = ?
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F(1, a) = 1 + a^2 - 4a + 2a - 4 = 0 ⇒ a^2 - 2a - 3 = 0 ⇒ a = -1 V a = 3
F '(x , y) = -(∂F / ∂x) / (∂F / ∂y) = -(3x^2 + y^2 - 4y) / (2xy - 4x + 2)
F '(1 , -1) = -[ 3·(1)^2 + (-1)^2 - 4(-1) ] / [ 2·1·(-1) - 4·1 + 2 ] = -(3 + 1 + 4) / (-2 - 4 + 2) = -8 / (-4) = 2
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Notlar:
F(x, y) = c
(∂F / ∂x)·(∂x / ∂x) + (∂F / ∂y)·(∂y / ∂x) = 0
(∂F / ∂x)·1 + (∂F / ∂y)·(dy / dx) = 0
dy / dx = F '(x, y) = -(∂F / ∂x) / (∂F / ∂y)
http://www.math.ucla.edu/~archristian/teaching/32a-f16/implicit-function-theorem.pdf (sayfa 2)

İkinci Türev Formülü:

https://math.stackexchange.com/questions/1935669/implicit-function-theorem-in-bbb-r2