Date: 201907
Local time of day: 0601-1200
Location: ZZZ.Airport, US
Flight conditions: IMC
Weather elements / visibility: Thunderstorm
Aircraft: B737-800
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Takeoff / Launch
ATC / advisory: Tower ZZZ
Aircraft component: Hydraulic System Pump
Component problem: Malfunctioning
Reporter organization: Air Carrier
Reporter function: Pilot Not Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Workload; Human-Machine Interface; Fatigue; Communication Breakdown; Time Pressure
Second reporter function: First Officer; Pilot Flying
Anomaly: Aircraft Equipment Problem Less Severe; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Flight Crew
When detected: In-flight
Result: Flight Crew Overcame Equipment Problem
Contributing factors: Aircraft; Human Factors
Primary problem: Human Factors

Narrative: Cleared to line up and hold for Runway XXL; the FO (First Officer) stated I have an amber light. I told him to get the QRH while I tell Tower. Tower cleared us off the active runway and to circle back around and hold short of the runway while we sort through our problem. FO stated we had an ELEC2 hydraulic low pressure light and the procedure is to turn the pump off; end of checklist. Tower told all departing flights that thunderstorms may shutdown south departure flows. Tower asked if we were going to be delayed long. I told the FO I was ready. He asked if I was willing to takeoff and I said yes without realizing I had not gone through the MEL flow chart. After takeoff and clear of the thunderstorms; I noticed the amber Low Pressure light on the overhead panel and realized in my haste to get airborne; I forgot the MEL flow chart process.Three-day trip and I was reassigned on day one. I arrived at hotel at XA:00 and didn't get a good nights sleep; but didn't think I was fatigued. Day two; ground stop to ZZZ; but we were able to land at ZZZ only 30 minutes late. The FAs (Flight Attendants) informed me we had international connections. As soon as gate was open; we taxied to the gate; but agent had difficulty with jet bridge. As the FO and I walked past the gate for [the connecting flight]; we saw our passengers at the gate unable to board as the door was shut 10 minutes before departure. We were both upset we were not able to get them to their connection. This lead to a less than restful night sleep for me thinking what I could have done better. Nevertheless; I was tired on day three but still did not think I was fatigued. Uneventful taxi out for takeoff to ZZZ1. [Weather service] displayed a line of thunderstorms approaching ZZZ. Seven aircraft ahead of us and Tower informed all aircraft they may ground stop south departures. Finally; we had clearance to line up and wait and I felt we were going to make it. Then the Low Pressure light came on. The FO ran the QRH and I thought we were ready. I was so focused on getting airborne before the thunderstorms shut down the departures; I simply forgot about the MEL. I believe being tired and frustrated coupled with my desire to get airborne before the thunderstorms shutdown the airport caused my focus to be misguided.As I write this report; I ask myself how could I have simply forgotten to run the MEL? It just doesn't make sense to me now. But at that exact moment; it seemed so clear. The QRH checklist was called complete and we were good to go. We going to be able to get airborne and get these passengers to ZZZ1 on-time. My lesson learned here is to maintain my focus on procedures.


Second narrative: [Report narrative contained no additional information.]


Synopsis: B737-800 flight crew reported rushed departure due to impending weather which resulted in a failure to complete MEL checklist.