Question

How do you differentiate `y=x^sqrt(x)`?

Answer 1

`(dy)/(dx)=x^{sqrt(x)}( 1/(2 sqrt(x))log(x)+sqrt(x)/x)`

Explanation:

Taking a easy route:
Applying the logarithmic transformation to both sides
`log(y) = sqrt(x)log(x)`
deriving now
`dy/y = 1/(2 sqrt(x))log(x)dx+sqrt(x)/xdx`
so
`(dy)/(dx) = y( 1/(2 sqrt(x))log(x)+sqrt(x)/x)`
and finally
`(dy)/(dx)=x^{sqrt(x)}( 1/(2 sqrt(x))log(x)+sqrt(x)/x)`