If y=x+e^x then d^2 x d y^2 =

Step-by-step solution

Given y = x + e^x . Differentiating both sides with respect to x , we get dy dx = 1 + e^x . Therefore, dx dy = 1 dy dx = 1 1 + e^x = (1 + e^x)⁻¹ . Now, differentiating dx dy with respect to y using the chain rule: d^2x dy^2 = d dy ((1 + e^x)⁻¹) = d dx ((1 + e^x)⁻¹) dx dy . d^2x dy^2 = -1(1 + e^x)⁻² e^x dx dy . Substituting dx dy = 1 1 + e^x : d^2x dy^2 = -e^x(1 + e^x)⁻² 1 1 + e^x = -e^x (1 + e^x)^3 . Answer: -e^x (1+e^x )^3