If x^x + y^x = 1, find

Let x^x = u and y^x = v

Taking log on both sides we get,

x log x = log u ……(1),

x log y = log v ……(2)

Using

Differentiating both sides of equation (1) we get,

Differentiating both sides of equation (2) we get,

We know that, from question,

u + v = 1

Differentiating both sides we get,

Putinng the value of eq(4) and eq(5) in equation above we get,