Find y'' by implicit differentiation. x² + 7y² = 7

Solution:

Given x² + 7y² = 7

Using implicit differentiation,

2x. dx/dx + 7 2ydy/dx = 0

2x + 14y dy/dx = 0

From further differentiation, we get

2 +14(dy/dx. dy/dx + y.d²y/dx²) = 0

2+ 14((dy/dx)² + y. d²y/dx²) = 0

14((dy/dx)² + y. d²y/dx²) = -2

((dy/dx)² + y. d²y/dx²) = -1/7

(y. d²y/dx²) = -1/7 - (dy/dx)²

d²y/dx² =  [-1/7 - (dy/dx)²] / y

Hence y'' = [-1/7 - (dy/dx)²] / y

Find y'' by implicit differentiation. x² + 7y² = 7

Summary:

y'' by implicit differentiation of x² + 7y² = 7 is  [-1/7 - (dy/dx)²] / y