If e^y + xy = e , then the ordered pair ( d y d x , d ^2 y d x^2 ) at x = 0 is equal to

Step-by-step solution

Given equation is e^y + xy = e . Substituting x = 0 in the given equation: e^y + 0 = e y = 1 Differentiating the given equation with respect to x : e^y d y d x + y + x d y d x = 0 Substituting x = 0 and y = 1 : e^1 d y d x + 1 + 0 = 0 d y d x = -1 e Differentiating again with respect to x : e^y ( d y d x )^2 + e^y d ^2 y d x^2 + d y d x + d y d x + x d ^2 y d x^2 = 0 e^y ( d y d x )^2 + 2 d y d x + (e^y + x) d ^2 y d x^2 = 0 Substituting x = 0 , y = 1 , and d y d x = -1 e : e ( -1 e )^2 + 2 ( -1 e ) + (e + 0) d ^2