Question

Differentiate \[x^{1/x}\]  with respect to x.

Answer 1

Let `y=x^(1/x)`

Take the natural logarithm of both sides: `log y = 1/x log x`

Differentiating log y gives:

`1/y dy/dx = d/dx (log x/x)`

`d/dx (log x/x)=((1/x)xxx-logxxx1)/x^2`

`1/y dy/dx = (1-logx)/x^2`

Multiply through by y to isolate `dy/dx`

`dy/dx = yxx (1-logx)/x^2`

Substitute y = x^1/x back into the equation:

`dy/dx = x^(1/x) xx (1-logx)/x^2`