According to the definition of the bilateral Laplace Transform: $$ X(s)=\int_{-\infty}^{+\infty}x(t)e^{-st}dt$$ where $s=\sigma+i\omega$. So to get the derivative property, $y(t)=\frac{dx(t)}{dt}$: $$Y(s)=\int_{-\infty}^{+\infty}\frac{dx(t)}{dt}e^{-st}dt $$ and using integration by parts we get: $$ Y(s)= x(t)e^{-st}\bigr|_{-\infty}^{+\infty} \ -\int_{-\infty}^{+\infty}-se^{-st}x(t)dt \ =x(t)e^{-st}\bigr|_{-\infty}^{+\infty} +sX(s) $$
Question: How can one get rid of the first term? Since the correct answer is just the other term.