Suppose that $f(x)$ is a continuous function on $[0,+\infty)$. Suppose that the improper integrals $\int_0^{+\infty}xf(x)dx$ and $\int_0^{+\infty}\frac{f(x)}{x}dx$ converge. Show that the improper integral $\int_0^{+\infty}x^tf(x)dx$ is defined for $t\in(-1,1)$ and it is continuously differentiable for $t\in(-1,1)$.
By abelian criterion we can know that the improper integral is convergent. But I could not prove the differentiablity. Naturally I guess the derivative is $\int_0^{+\infty}x^t\ln xf(x)dx$, whose convergence can be proved by abelian criterion.