Much longer comment turned answer:
Howard Eves' An Introduction to the History of Mathematics seems like a perfect fit. The chapters that would be of particular use are chapters 11 through 14 (relevant sections included on the side):
- Chapter 11: The Calculus and Related Concepts [11.9 Newton; 11.10 Leibniz]
- Chapter 12: The Eighteenth Century and the Exploitation of the Calculus
- Chapter 13: The Early Nineteenth Century and the Liberation of Geometry and Algebra [13.5 Cauchy]
- Chapter 14: The Later Nineteenth Century and the Arithmetization of Analysis [14.10 Weierstrass and Riemann]
Of course, there are many other math history books that may be suitable but Eves' is a classic. It's very well-written, non-technical, and there are problems at the end of each chapter that address historical points made in the text.
It should probably be noted that analysis is almost inherently a technical subject; thus, you may grab a book different than Eves' with more depth, but what you make up in depth will probably be lost in increasing technicality (if that made sense).