What is derivative of `sqrt(x^x)`?
Let, `y = sqrt(x^x)=x^(x^(1/2))`
`rArr y= x^(1/2*x)` [since `u^(v^z)=u^(v*z)` ]
Taking log of both sides,
`lny = ln(x^(1/2*x)) =(1/2)xlnx` [since `ln(u^v)=vlnu` ]
For differentiation, the left hand side of the equation requires chain rule whereas the right hand side requires product rule. Thus,
Multiplying both sides by y, to eliminate it from the L.H.S., we get
Therefore, the derivative of `sqrt(x^x)` is `1/2*sqrt(x^x)(lnx+1)` (assuming, x>0)