Possible derivation:
d/dx(x^x)
Express x^x as a power of e: x^x = e^(log(x^x)) = e^(x log(x)):
= d/dx(e^(x log(x)))
Using the chain rule, d/dx(e^(x log(x))) = ( d e^u)/( du) 0, where u = x log(x) and ( d)/( du)(e^u) = e^u:
= (d/dx(x log(x))) e^(x log(x))
Express e^(x log(x)) as a power of x: e^(x log(x)) = e^(log(x^x)) = x^x:
= x^x d/dx(x log(x))
Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x and v = log(x):
= log(x) d/dx(x)+x d/dx(log(x)) x^x
The derivative of x is 1:
= x^x (x (d/dx(log(x)))+1 log(x))
The derivative of log(x) is 1/x:
= x^x (log(x)+1/x x)
Simplify the expression:
Answer: |= x^x (1+log(x))
