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Calculus is built on two operations — differentiation, which is used to analyse instantaneous rate of change, and integration, which is used to analyse areas. Understanding differentiation and using it compute derivatives of functions is one of the main aims of this course.

We had a glimpse of derivatives in the previous chapter on limits — in particular Sections 1.1 and 1.2 on tangents and velocities introduced derivatives in disguise. One of the main reasons that we teach limits is to understand derivatives. Fortunately, as we shall see, while one does need to understand limits in order to correctly understand derivatives, one does not need the full machinery of limits in order to compute and work with derivatives. The other main part of calculus, integration, we (mostly) leave until a later course.

The derivative finds many applications in many different areas of the sciences. Indeed the reason that calculus is taken by so many university students is so that they may then use the ideas both in subsequent mathematics courses and in other fields. In almost any field in which you study quantitative data you can find calculus lurking somewhere nearby.

Its development  1  came about over a very long time, starting with the ancient Greek geometers. Indian, Persian and Arab mathematicians made significant contributions from around the \(6^{\rm th}\) century. But modern calculus really starts with Newton and Leibniz in the \(17^{\rm th}\) century who developed independently based on ideas of others including Descartes. Newton applied his work to many physical problems (including orbits of moons and planets) but didn't publish his work. When Leibniz subsequently published his “calculus”, Newton accused him of plagiarism — this caused a huge rift between British and continental-European mathematicians which wasn't closed for another century.