This is kind of off-topic but also applies to some extent
Changing up when you get out of the saddle is not "exactly what you should do", the situation you're in dictates what is the appropriate response. Certainly one of these responses is changing up a gear or two but equally it may also need you to drop a gear or even a chainring!
Changing up a gear or two is fine & dandy if you're okay to comfortable in the saddle but are wanting to accelerate & if you're transiting onto a steeper section of the climb. However if you're transitioning to a much steeper climb you'll want to keep in the gear or even drop a gear as you stand up. Worse than this if you're standing up because you're about to stall changing up a gear or two will only make the stall happen & so you drop a gear or a chanring if you have any to spare & accelerate changing up when you get to a slow sprint cadence & possibly when you sit down. Learning how to drive through from a climbing out of the saddle cadence to a sprinting cadence is difficult one at first, as most riders find there's a cadence & technique gap between the two, but once mastered very effective at the top of a hill with a steep kick at the top especially if you can step up to the top chainring.
Fair points - I was referring to "for a constant gradient" I needed to be in a harder gear - I think it is because I have more power available standing up, but obviously I can't keep that intensity up for as long.