Since the given function is a quotient, we must use the quotient rule:

d

dx
ex ex

ex + ex

=
(ex+ex) (d/dx)(exex) (exex)(d/dx)(ex+ex)

(ex+ex)2
=
(ex+ex)(ex+ex) (exex)(exex)

(ex+ex)2
(since the derivative of ex is ex)
=
(e2x+2exex+e2x) (e2x2exex+e2x)

(ex+ex)2
(expanding the terms)
=
2exex + 2exex

(ex+ex)2
(because the other terms cancel)
=
4

(ex+ex)2
(because exex = 1)

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