If x^2 a^2 + y ^2 ~b ^2 =1 , then d ^2 y d x^2 is

Step-by-step solution

Implicit differentiation of the equation x^2 a^2 + y^2 b^2 =1 yields: 2x a^2 + 2y b^2 dy dx = 0 Solving for the first derivative gives dy dx = - b^2x a^2y . Differentiating again using the quotient rule: d^2y dx^2 = - 1 a^2 ( y b^2 - b^2x dy dx y^2 ) Substituting the expression for dy dx produces: d^2y dx^2 = - 1 a^2 ( b^2y + b^4x^2 a^2y y^2 ) = - b^2(a^2y^2 + b^2x^2) a^4y^3 The original equation implies b^2x^2 + a^2y^2 = a^2b^2 , which simplifies the expression to: d^2y dx^2 = - b^4 a^2y^3 This corresponds to opti