\phi '\left( x \right) = \int_0^1 {\frac{{\partial f}}
{{\partial x}}\left( {x,t} \right)dt} = \int_0^1 {t^x dt} = \left. {\frac{{t^{x + 1} }}
{{x + 1}}} \right|_0^1 = \frac{1}
{{x + 1}}