The symbol dx means different things depending on the context. In calculus. it's commonly used to denote a small increment in x. It can also stand for a more abstract object called an infinitesimal. It can also stand for a 1-form which is a linear functional acting on the tangent space of a differentiable manifold.
dy/dx represents a function of x, called the derivative function. It takes an element of the domain of the function y = f(x) for which f is differentiable and outputs the "instantaneous" rate of change of y with respect to x. You can think of this as outputting the slope of the tangent line to the curve at the point (x, f(x)).
Exciting news! I’m working through Lang’s Basic Mathematics but already own the Geometry book!
The Geometry book has a separate solutions manual.
Merry Christmas 🎄
Great video! Will you finish the playlist on the book by Shifrin? I was really enjoying it.
On this topic, how is Shifrin's book Linear Algebra a Geometric Approach? Does anyone know
Yes, finishing Shifrin and Simmons is on the docket for after I finish Spivak and Tao. After that, I'll finish Axler and Taylor.
Grat to hear, I am also looking forward to Axler. Thanks for the reply!@@learningasahobby790
Where did you find a hardcopy of Lang's Basic Mathematics?
It's a used copy. Unfortunately, I don't remember where I got it from. I've had it for many years.
Professor, does dx really mean a very small change in x or can it be interpreted as for every x?
The symbol dx means different things depending on the context. In calculus. it's commonly used to denote a small increment in x. It can also stand for a more abstract object called an infinitesimal. It can also stand for a 1-form which is a linear functional acting on the tangent space of a differentiable manifold.
My professor told me that the interpretation of dy/dx is the change in y for every x. I want to know whether this is mathematically correct.

dy/dx represents a function of x, called the derivative function. It takes an element of the domain of the function y = f(x) for which f is differentiable and outputs the "instantaneous" rate of change of y with respect to x. You can think of this as outputting the slope of the tangent line to the curve at the point (x, f(x)).