We give the definition of derivatives and prove standard differentiation rules that most of us learned in an introductory calculus class in Sect. 5.1. One major difference in my approach to this compared to many others is the explanation of \(\frac{d}{dx}e^{x} = e^{x}\) . We derived \(e^{x} = \mathop \sum \limits_{n = 0}^{\infty } \frac{{x^{n} }}{n!}\) in Sect. 4.4. We will use it as a plausible explanation of \(\frac{d}{dx}e^{x} = e^{x}\) . However, this is only a plausible explanation at this point. The conclusive explanation is postponed to Chap. 7. In Sect. 5.2, we prove the mean value theorem and Cauchy’s mean value theorem as consequences of the Rolle’s theorem. Then we prove L’Hospital’s rule and the increasing/decreasing theorem. We give applications of these theorems, and end with anti-derivatives. Section 5.3 is on continuous but nowhere differentiable functions.