Expert: Ahmed Salami - 12/11/2009

Question
find d^2/dx^2 by implicit differentiation for the equation: y^2=1-2/X

Answer
Hi Kelly,
y² = 1 - 2/x
differentiating implicitly
2y(dy/dx) = 2/x²
y(dy/dx) = 1/x²
differentiating again
[y(d²y/dx²) + (dy/dx)(dy/dx)] = -2/x³
y(d²y/dx²) + (dy/dx)² = -2/x³
y(d²y/dx²) = -2/x³ - (dy/dx)²
d²y/dx² = -[2/x³ + (dy/dx)²]/y
       = -[2 + x³(dy/dx)²]/x³y
but dy/dx = 1/x²y
(dy/dx)² = 1/(x^4)y²
d²y/dx² = -[2 + x³/(x^4)y²]/x³y
       = -[2 + (1/xy²)]/x³y
       = -(2xy² + 1)/(x^4)y³

Regards