Question

Differentiate y=x^√x

Answer 1

Rewrite the right hand side in terms of exponentials and logarithms, using the fact that


`a=e^(\ln a)=\exp(\ln a)`


`y=x^(\sqrt x)=\exp\left(\ln x^(\sqrt x)\right)=\exp\left(\sqrt x\ln x\right)`

Now differentiate both sides, applying the chain and product rules.


`(\mathrm dy)/(\mathrm dx)=(\mathrm d)/(\mathrm dx)\exp\left(\sqrt x\ln x\right)`


`(\mathrm dy)/(\mathrm dx)=\exp\left(\sqrt x\ln x\right)\cdot(\mathrm d)/(\mathrm dx)\left(\sqrt x\ln x\right)`


`(\mathrm dy)/(\mathrm dx)=x^(\sqrt x)\left((\mathrm d)/(\mathrm dx)[\sqrt x]\ln x+\sqrt x(\mathrm d)/(\mathrm dx)[\ln x]\right)`


`(\mathrm dy)/(\mathrm dx)=x^(\sqrt x)\left((\ln x)/(2\sqrt x)+\frac{\sqrt x}x\right)`


`(\mathrm dy)/(\mathrm dx)=\frac12x^(\sqrt x)\left(x^(-1/2)\ln x+2x^(-1/2)\right)`


`(\mathrm dy)/(\mathrm dx)=\frac12x^(\sqrt x-1/2)\left(\ln x+2\right)`