Derivative of x^ (x^x ) is

Step-by-step solution

Let y = x^ (x^x) . Applying logarithmic differentiation, take logarithms on both sides: y = (x^ (x^x) ) = x^x x Differentiating both sides with respect to x : 1 y dy dx = d dx (x^x x) To differentiate x^x x , first find d dx (x^x) by letting u = x^x : u = x x Differentiating: 1 u du dx = x + 1 Thus du dx = x^x(1 + x) Now apply the product rule to x^x x : d dx (x^x x) = x^x(1 + x) x + x^x 1 x = x^x x (1 + x) + x^ x-1 Substituting back: 1 y dy dx = x^x x (1 + x) + x^ x-1 Solving for the derivative: dy dx = x^ (x^x) (