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Jol had come under fire after Spurs lost their opening two games which left them rooted at the foot of the Premiership.
But two goals from Malbranque inside the opening six minutes quickly eased any fears he could be the first manager to be given the sack this season.
A superb strike from Jermaine Jenas and Darren Bent's first goal for the club ensured Tottenham moved out of the bottom three and left Derby facing a long hard season in the top-flight.
Jol saw his side get off to a dream start when Malbranque opened the scoring after just 74 seconds.
Jenas played a short free-kick into the path of the former Fulham midfielder and he unleashed a powerful shot from the edge of the box which flew past Stephen Bywater.
Things were to get even better for Jol when the Frenchman added a second for the home side on six minutes.
Darren Bent laid the ball across to Malbranque who hit a right-foot shot past Bywater into the corner of the net
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Jenas was determined to prove a point to the Tottenham fans after he was booed by sections of the crowd during the home defeat against Everton last Tuesday.
And the England midfielder responded in the best possible manner by getting his name on the scoresheet and putting the result beyond Derby on 14 minutes.
Jenas took the ball off Andy Todd and burst his way towards the Derby goal before sliding the ball past Bywater for his first goal of the season.
Just when Derby thought it could not get any worse, Wayne Routledge had the ball in the back of the net on 32 minutes.
Robbie Keane's mis-hit fell straight to the talented winger and he made no mistake firing the ball home, but his effort was ruled out for offside.
It was all one-way traffic and Derby could not handle the attacking dominance of the home side.
Jenas should have grabbed his second of the game on 36 minutes when his curling shot flew inches past the post, while Bent had strong claims for a penalty turned down on 42 minutes when he appeared to be brought down inside the area by Jay McEveley but referee Chris Foy waved for play to continue.
Derby could have grabbed a consolation on 70 minutes but Rams substitute Gary Teale saw his shot bring a superb save from Paul Robinson in the Tottenham goal.
Bent finally scored his first goal for Tottenham when he added a fourth for the home side on 79 minutes.
McEveley's back header produced a fine save from Bywater, but the rebound fell straight to the England striker, who made no mistake heading the ball home from close-range.