- #1
madness
- 815
- 70
I can't help but feeling these days that I don't actually understand where most of the maths I use comes from. Unfortunately, I can't remember whether this is due to the fact that I didn't take my studies seriously until the end of undergrad, or rather that these things were never actually taught to me. One example of this is the derivative of elementary functions that I use regularly (things like trig functions, exponents, etc.). Let's take the exponential function. It's easy to show from the definition of a derivative that $$\frac{d a^x}{dx} = a^x \lim_{h\rightarrow 0}\frac{a^h -1}{h}$$ (at least for) $$a\ne0$$. However I don't know how to take that limit to complete the proof. So my concrete question is: can one actually complete this proof in a straightforward way, that is without other specialised knowledge of the function (e.g., its power series definition, relationship to logarithm, etc.) I would be in the same situation with all the common functions, and this situation case extends beyond calculus to other fields such as linear algebra. In linear algebra for example, it struck me from reading another thread here that to prove that matrices have at least one eigenvalue, one must use the fundamental algebra, whose proof I didn't see until I took algebraic topology. So my more general question is: do we (most of us?) typically learn mathematics procedurally/operationally and never really know the reasons why the things we do are actually valid? Or is it just me?
PS: I'm aware that I've interspersed a number of different questions and points here, I hope that doesn't cause umbrage with anyone!
PS: I'm aware that I've interspersed a number of different questions and points here, I hope that doesn't cause umbrage with anyone!