If y = x^x , then the value of ( dy dx )^2 at x = e^2 is

Step-by-step solution

Given y = x^x Using the property of logarithms, this can be written as: y = x x Differentiating both sides with respect to x using the product rule: dy dx = x 1 x + x 1 dy dx = 1 + x Substituting x = e^2 into the derivative: dy dx = 1 + (e^2) dy dx = 1 + 2 e Since e = 1 , we get: dy dx = 1 + 2 = 3 Therefore, the value of ( dy dx )^2 is: ( dy dx )^2 = 3^2 = 9 Answer: 9