Es ist

${\displaystyle {}{\frac {\partial }{\partial t}}{\left(\sin \left(\lambda x\right)e^{-\lambda ^{2}t}\right)}=-\lambda ^{2}\sin \left(\lambda x\right)e^{-\lambda ^{2}t}\,}$

und

${\displaystyle {}{\frac {\partial }{\partial x}}{\left({\frac {\partial }{\partial x}}{\left(\sin \left(\lambda x\right)e^{-\lambda ^{2}t}\right)}\right)}={\frac {\partial }{\partial x}}{\left(\lambda \cos \left(\lambda x\right)e^{-\lambda ^{2}t}\right)}=-\lambda ^{2}\sin \left(\lambda x\right)e^{-\lambda ^{2}t}\,,}$