Find (d^2)y/d(x^2) at the following point using implicit differentiation: sqrt(x)+sqrt(y)=1, at (1,0)

Answer 1

x^1/2 + y^1/2 = 1

1/2 x ^-1/2 + 1/2 y^-1/2 y' = 0

y' = - x^-1/2 / y^-1/2 = - (y/x)^1/2

y'' = - 1/2 (y/x) ^-1/2 ((1. y' - y)/x²) using the power and quotient rules

You can substitute in for y'

Answer 2

√x + √y = x^1/2+y^1/2 = 1

First derivative:

(x^-1/2/2)dx + (y^-1/2/2)dy = 0

dy/dx = - √(y/x) = -y^1/2/x^1/2

d²y/dx² = -[(y^-1/2x^-1/2/2)dy/dx - y^1/2x^-3/2/2]

= -[(y^-1/2x^-1/2/2)(-y^1/2x^-1/2) - y^1/2x^-3/2/2]

= -[(1/2)(-1/x) - y^1/2x^-3/2/2]

= 1/(2x) +√y/(2x^3/2)

= (√x+√y)/(2x^3/2))

d²y/dx²(1,0) = 1/2