What is Differential calculus?
'''Differential calculus''', a branch of [[calculus]], is the study of finding out the rate of change of a [[variable]] compared to another variable, by using [[function (mathematics)|functions]]. It is a way to find out how a shape changes from one point to the next, without needing to divide the shape into an [[infinite]] number of pieces. Differential calculus is the opposite of [[integral calculus]]. It was developed in the 1670s and 1680s by Sir [[Isaac Newton]] and [[Gottfried Leibniz]]. ==Background== Unlike a number such as 5 or 200, a variable can change its value. For example, [[distance]] and [[time]] are variables. At an Olympic running race, as the person runs, their ''distance'' from the starting line goes up. Meanwhile, a [[stopwatch]] or clock measures the ''time'' as it goes up. We can measure the average [[speed]] of the runner, if we divide the distance they travelled by the time it took. But this does not say what speed the person was running at exactly 1.5&nbsp;seconds into the race. If we had the distance at 1&nbsp;second and the distance at 2&nbsp;seconds, we would still only have an average, although it would probably be more correct than the average for the whole race. Until calculus was invented, the only way to work this out was to cut the time into smaller and smaller pieces, so the average speed over the smaller time would get closer and closer to the actual speed at exactly 1.5&nbsp;seconds. This was a very long and hard process, and had to be done each time people wanted to work something out. Certainly it is much more difficult for a driver to figure out a car's speed by using only its odometer (distance meter) and clock—without a speedometer. [[File:Simple curve showing tangents.svg|thumb|right|On a curve, two different points have different slopes. The red and blue lines are [[Trigonometric function|tangent]]s to the curve.]] A very similar problem is to find the [[wikt:slope|slope]] (how steep it is) at any point on a [[curve]]. The slope of a ''[[straight]]'' line is easy to