Let y=xeˣ then:
ln(y)=exln(x); differentiating:
(1/y)(dy/dx)=ex/x+exln(x); differentiating again:
(-1/y2)(dy/dx)2+(1/y)(d2y/dx2)=2ex/x-ex/x2+exln(x),
(1/y)(d2y/dx2)=2ex/x-ex/x2+exln(x)+(1/y2)(dy/dx)2,
(1/y)(d2y/dx2)=2ex/x-ex/x2+exln(x)+(ex/x+exln(x))2,
d2y/dx2=y(2ex/x-ex/x2+exln(x)+(ex/x+exln(x))2),
d2y/dx2=yex(2/x-1/x2+ln(x)+ex(1/x+ln(x))2),
d2y/dx2=yex(2/x-1/x2+ln(x)+ex(1/x2+2ln(x)/x+ln2(x))),
d2y/dx2=xeˣex(2/x-1/x2+ln(x)+ex(1/x2+2ln(x)/x+ln2(x))).
There are several ways to write this. For example, xeˣ/x=xeˣ-1, etc.