If y = x^2 + 8x + 3 , then the value of d^2 y dx^2 at x = 3 is

Step-by-step solution

Given y = x^2 + 8x + 3 Squaring both sides: y^2 = x^2 + 8x + 3 Differentiating with respect to x : 2y dy dx = 2x + 8 y dy dx = x + 4 Differentiating again with respect to x : y d^2 y dx^2 + ( dy dx )^2 = 1 At x = 3 , y = 3^2 + 8(3) + 3 = 36 = 6 Substituting x = 3 and y = 6 in the first derivative equation: 6 dy dx = 3 + 4 dy dx = 7 6 Substituting y = 6 and dy dx = 7 6 in the second derivative equation: 6 d^2 y dx^2 + ( 7 6 )^2 = 1 6 d^2 y dx^2 = 1 - 49 36 = -13 36 d^2 y dx^2 = -13 216 Answer: -13 216