Differences Between Après vs. Derrière and Avant vs. Devant

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Après and Avant convey a notion of time or space. Après refers to doing something after while Avant refers to doing something before.

Je le retrouve après/avant le déjeuner
I'll meet up with him after/before lunch
Après/avant le bois, il y a un chemin
After/before the wood, there is a path

Derriere and Devant convey a notion of precise space. Derrier refers to being behind something, or someone and Devant refers to being in front of something or someone.

La petite fille est cachée derrière l'arbre
The young girl is hidden behind the tree
Pour la photo, comme tu es plus petite, va devant Camille.
For the picture, since you are smaller, go in front of Camille. 
Derrière le bois, il y a un chemin
Behind the wood, there is a path

Après and Derrière Are Not Interchangeable

So, what is the difference between the two sentences "après le bois, il y a un chemin" and "derrière le bois, il y a un chemin"?

They both give a piece of space-related information, but one is more precise, just like in English. Same exact logic applies to avant versus devant.

Après Que + Indicative / Avant Que + Subjunctive

A common mistake is Après que plus a subjunctive. It's a very common mistake, even among French people, because honestly, the indicative sounds terrible there. Avant que is followed by the subjunctive because we don't yet know if the action is going to become a reality. With Après que, the action has taken place already: there is no doubt left, hence no need for the subjunctive.

Après que + subjunctive sounds so bad to a French ear that we will do our best to use a noun instead of a verb after. You can use the same trick with "avant que" et avoid using a subjunctive.

Je dois commencer après qu'il part. (or après son départ)
I must start after he leaves (or after his departure).
Je dois commencer avant qu'il parte (or avant son départ).
I must start before he leaves (or before his departure)

By the way, even if we use "le derrière" in French (although this is extremely polite, just like saying "the behind" in English), French people use the preposition "derrière" without thinking about it at all. Just like in English you use "behind" without thinking about that part of the anatomy.