Implicit differentiation－hzb53jl55v的部落格

標題:

Implicit differentiation

發問:

Find (d^2) y / d (x^2) given X^2+ y^2 = 1

最佳解答:

x^2 + y^2 = 1 First-order derivative: 2x + 2y(dy/dx) = 0 2y(dy/dx) = -2x dy/dx = (-2x)/(2y) dy/dx = -x/y Second-order derivative: Using quotient rule, (dy/dx)^2 = [y(-1) - (-x)(dy/dx)] / y^2 Sub dy/dx = -x/y, = [-y + x(-x/y)] / y^2 = [-y - x^2/y] / y^2 = [(-y^2 - x^2) / y] / y^2 = -(x^2 + y^2) / y^3 Sub x^2 + y^2 = 1, (dy/dx)^2 = -1/y^3