ABSTRACT
This chapter looks at a few different ideas from calculus. It remarks that though the concept of limit is central to all of calculus. The chapter assumes that any generic functions students studies are continuous and differentiable on their domains. One can use the Quotient Rule to establish the Power Rule for negative integers, then use implicit differentiation to establish it for all rational numbers. The first of these is by far the more “popular” since it provides a very quick way to compute the exact value of an integral provided that students can write down an anti-derivative of your given function, which may or may not be possible. Its proof depends upon the Mean Value Theorem and, though not too complicated, is a bit messy, so students can skip it here.
