My book Calculus from the Ground Up focuses on differentials, and uses it to provide a unification of process and simplification of understanding of a lot of different parts of calculus.
To read about the thought process that led to the book you can see this arXiv link; the focus on differentials that you are asking for led naturally to a refactoring of the way introductory calculus is presented.
Differences from other books:
The arXiv link gives some important information, but I'll repeat some of it here. First of all, the focus of the entire book is on differentials. We do a lot of derivatives, but the focus is always on differentials, and for several important reasons. First, it unifies several important practices into a single system - single-variable, multivariable, and implicit differentiation all has the exact same process. Second, it makes the different geometric integrals more obvious. The integral is presented as a sum of infinitesimals, not as an area under the curve (which becomes merely one of the application areas). The integral simply sums up whichever geometry is being used. $\int y\,dx$ for summing areas of rectangles, $\int \pi y^2\,dx$ for summing volumes of cylinders, and $\int \sqrt{dx^2 + dy^2}$ for summing arc lengths. The way you are asked to memorize them is exactly what the geometry states. For instance, many books want you to have the volume of cylinders as $\pi \int y^2\,dx$. That's correct, but moving the $\pi$ outside means that it no longer looks like the volume of a cylinder equation for students.
Additionally, the book includes a rule that seems to have gone missing for doing differentials of the form $u^v$. For those who don't know (because it is missing in most modern books), $d(u^v) = vu^{v - 1}du + \ln(u)u^vdv$ (I wish the font for $v$ had a more distinct look here, but oh well). Many books teach "logarithmic differentiation" for this, but it is wholly unnecessary. Just like all the other differentials, all you need is the rule.
I also try to include additional life lessons that we can learn from calculus. For instance, in the discussion of Taylor polynomials, I discuss how this can be used as a template for solving impossible problems (not only in math but anywhere).
Also, I wanted to make a note on the second differential, because it came up in the discussion of Keisler. I don't make a big deal about it (I put it in the Appendix), but I actually introduce a form for the second derivative that makes the chain rule for the second derivative work algebraically. Generally, in the text, I avoid this situation by simply introducing a variable for the first derivative, and then take the derivative of that variable. However, in the appendix I show that second differentials can be made algebraic by making the second derivative $\frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2}$. If that looks strange to you, you can derive it for yourself by simply taking the derivative of $\frac{dy}{dx}$. Note that $\frac{dy}{dx}$ is a quotient, so you would use the quotient rule to take the derivative of it. This leads to differentials that are 100% algebraically manipulable. Most texts focusing on differentials don't tell you either the problem nor the solution for using second differentials.
The structure of the book differs from "Calculus Made Easy" in that it starts with derivatives, since a slope is more intuitive for people coming from algebra. Unlike Keisler, it saves discussion of limits for the end of the book. Essentially, it gives you the intuition and the toolset first, and then, at the end, goes into a bit more formally the underpinnings of what makes it work. I find that students prefer this approach. Like Keisler, I use the hyperreal numbers (though I don't formally introduce them until the last third of the book, which focuses on the infinite).
Anyway, I always try to write things in such a way as to focus the student on the intuitions behind everything, so that learning calculus doesn't just teach them calculus, but it improves their thinking. For instance, when talking about the other geometric uses of the integral (volumes from cylinders, volumes from shells, arc lengths, etc.), I gave a general mental mechanism that is used to generate all of these. (a) the problem can be estimated by a formula, (b) the problem can be divided into subproblems, (c) each subproblem must have the same form as (a), (d) the result must be attainable by adding the results of the subproblems, and (e) increasing the number of subdivisions improves the accuracy of the estimation method. The goal here is to show the students how the thought process works.
Also, my student's also love the fact that I show where all of their formulas that they learned in previous math classes come from. I show how to derive the interest rate formula, the volume of a cone formula, and the volume of a sphere formula. In fact, that's another aspect of the book - I teach how to derive formulas. We use calculus to derive the vertex formula for quadratics, and a homework problem is deriving the vertex formula for cubics. I tell students that calculus is the "where babies come from" of math.
NOTE - I edited this to include more details about the book and what makes it different because I was requested to below. Sorry if this comes off as more of an advertisement than was intended.
