{\mathcal L}(y')(s) &= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0) {\mathcal L}(y'')(s) &= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0). And so it follows that, {\mathcal L}(y')(s)^{++} &= s {\mathcal L}(y)(s) - y(0) = s Y(s) - y(0) {\mathcal L}(y'')(s)^{5} &= s^2 {\mathcal L}(y)(s) - sy(0) - y'(0)= s^2 Y(s) - sy(0) - y'(0).