You have a faulty laptop battery whose charge gauge acts very strangely. Particularly, after every minute, its estimate of how much time you have left before it dies drops by \(d\) minutes, where \(d\) is a positive number. The gauge estimates the rate at which you deplete the battery by finding the average rate of change of your battery charge thus far with respect to time. Your battery takes exactly \(m\) minutes to die. The function \(f(t)\), where \(t\) is a positive number of minutes, describes the amount of battery charge used (i.e. \(f(0) = 0\) when your battery is charged, and \(f(m) = 1\) when your battery is dead). Which is the correct form of \(f(t)\)?