Hi I'm reading Elementary calculus - an infinitesimal approach and just wan't to make sure my understanding of what dy, f'(x) and dx means is correct. I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the tangent to the graph which is the derivative of f at that point and tells you the slope of the function at that point. But I want the correct mathematical understanding of it. The book argues that Δy = the change in y along a curve between 2 points (I assume it's a secant) dy = change in y along the tangent line to that curve between 2 points (The differential) Then it defines that Δy = f(x+Δx)-f(x) dy = f'(x)Δx and mentiones that "let y=f(x). Suppose f'(x) exists at a cetain point x, and Δx is the infinitesimal, then Δy is infinitesimal and" Δy = f'(x)Δx+ εΔx and prooves that Δy/Δx ≈ f'(x) Δy/Δx = f'(x) + ε Δy = f'(x)Δx + εΔx My first question is This: Is f(x+Δx)-f(x) and f'(x)Δx+ εΔx Equal? If they both are equal to Δy then i assume they are? Is this the correct understanding: This expression Δy/Δx only approaches f'(x): Δy/Δx = (y2-y1)/(x2-x1) = (f'(x)Δx+εΔx)/((x+εΔx-x)) ≈ f'(x) While this expression is equal dy/dx = f'(x) The difference between dy/dx = f'(x) and Δy/Δx ≈ f'(x) is that Δy= dy+ εΔx contains that extra "εΔx" term and therefore is bigger than dy and is also the secant between 2 points along the graph, whereas dy is an infinitesimal small movement between 2 points on the tangent to the graph. Because dy=f'(x)*Δx equals the term f'(x)*Δx this tells us that dy is an infinitesimally small change in y between the point at the tangent and another point on the tangent infinitesimally close to that point. The Δx in f'(x)Δx shows us that it's a "change in y" corresponding to a infinitesimally small change in x so (x2-x1), hence Δx? If the f'(x) exists then the differential dy and the increment Δy MUST be infitesimal and so close together that they cannot be seen under the infinitesimal microscope. So my last question: I guess it's a no-go to treat the symbol dy/dx as a quotient? The book mentions that if dy=f'(x)dx and if dx ≠ 0 then we can rewrite the equation dy/dx = f'(x), is this just lucky that this "trick" works, or is it true that you can treat it like a quotient?