let $f\in L^1(0,\infty)$. For x>0, define $g(x)=\int_{0}^{\infty} f(t) e^{-tx} dt$. Prove that $f$ is differentiable for $ x>0$ and with derivative $g'(x) = \int_{0}^{\infty} -tf(t) e^{-tx} dt$.
To show g is differentiable, I tried to use sequence definition of derivatives and wanted to use dominated convergence theorem later I got stuck. Do you have any idea?. Any hints are admirable to this problem.