In this part we introduce the derivative of a real function. Its mastery is one of the key skills that one needs in mathematics, being able to differentiate fast and reliably is a necessary prerequisite for pretty much all math courses. We will try to help you along by trying to explain the process so that you grasp it intuitively (Theory, Methods Survey), then supply you with ample Exercises of varrying dificulty. If all goes well, by the time you are through here you should have a feeling that you can take a derivative of any function you encounter.
There are many theorems that use the knowledge of derivative to find out something about a given function. We dedicate one part to the Mean Value Theorem and related topics, perhaps the most important part of entry-level real analysis. In particular, derivative is used to determine monotonicity and concavity, so we dedicated one part to graphing function, where we also include practical look at topics covered in Functions (domain and limits, asymptotes) from the point of view of graph sketching.
In part Applications we look at tangent and normal lines, try our hand at simple optimization (the ever so popular word problems, basically a global extreme problem), and talk about approximatiot of functions, in particular the Taylor polynomial.
Since derivatives are so useful, all topics (apart from the more theoretical MVT sections) are accompanied by Methods Survey, Solved Problems and Exercises.