James McAtee's early strike proved to be the difference at the Academy Stadium, as a strong defensive performance ensured we took the three points and returned to the top of the PL2 table.
The scoreline was in stark comparison to some of our more goal heavy encounters so far this season.
But the EDS boss was pleased to see how the team coped with periods of relentless pressure applied by the visitors.
"It was a really tough game, a really brilliant game for us," said Barry-Murphy.
"I think it's very important for our players to understand different ways to win a game or to play the game.
"It's about understanding how good the opponent can be and how tough they can be and how prepared they were for this game.
"We had very little preparation time, so to see our players adapt so well and to fight until the very end for each other was all we could ask."
Saturday's clash marked the end of a very busy period for our EDS group, with UEFA Youth League and Papa John's Trophy fixtures adding to a hectic and demanding schedule.
It has taken it's toll on the squad with a number of players picking up injuries.
So Barry-Murphy was relieved to see his players come out of the Derby encounter unscathed.
"At the end of what has been a really challenging month, to see us come through without any injuries was big for me," the manager added.
"We have lost so many players over the course of this month through injury and pretty significant injuries.
"It has been a very productive month but when you lose players for what looks like significant time, it can be tough for the whole group."
Now some of our youngsters will head away on international duty with a number of our players named England's Under-17 to Under-21 squads.
And Barry-Murphy is pleased to see that his players will be exposed to a new type of challenge.
He added: "It is a brilliant opportunity for the lads to reflect.
"For the players to go and represent their countries, I think it is a brilliant feeling.
"The ones who stay here, we’ll look at fitness as we try and progress everyone to a situation where everybody feels that they are at the peak of their powers."