Proof of product rule and integration by parts

d(u*v)/dx = u*dv/dx + v*du/dx {?}
u*v = a + b
d(u*v)/dx = da/dx + db/dx
da = u*m = u*dv
db = v*n = v*du
d(u*v)/dx = u*dv/dx + v*du/dx
proof of product rule. Done. Extends easily!
u*v*h = a + b + c
v*h = l
b + c = f
u*l = a + f
d(u*l)/dx = da/dx + df/dx
da = u*m = u*dl
df = l*n = l*du
d(u*l)/dx = u*dl/dx + l*du/dx
and so,
d(u*v*h)/dx = u*d(v*h)/dx + v*h*du/dx
d(u*v*h)/dx = u*[vdh/dx + hdv/dx] + v*h*du/dx
d(u*v*h)/dx = u*v*dh/dx + u*h*dv/dx + v*h*du/dx

integration by parts
d(u*v)/dx = u*dv/dx + v*du/dx
~d(u*v) = ~u*dv + ~v*du
u*v = ~u*dv + ~v*du
u*v - ~u*dv = ~v*du