There is currently no inbuilt way to record game-play, in Destiny. The only way to record prior activity is through the Xbox One DVR. That said, you suggest that the Xbox might have a way to capture prior game play, of longer than 30 seconds. In actual fact, the standard DVR system has this mechanic in place.
To increase recording time, access Settings > All Settings > Preferences > Game DVR & streaming. Select the "Record that" records option, and select the time-span. While the default recording time is 30 seconds, you can choose to record anywhere between 15 seconds and 5 minutes. Anything longer, and you will simply have to make multiple recordings, and use video editing software to stitch them together.
To record the last 5 minutes, simply press on the guide button twice1, and hit X. You will return to your game, and the last 5 minutes of game-play will be saved as a recording.
1Pressing the guide button twice brings up the side menu, where you have quick access to things like your friend list, notifications and achievements. With this screen up, pressing Y will save a screenshot in-game, and pressing X will record the last 15 seconds - 5 minutes of game-play, depending on your allotted "Record that" time.