(Maths question) Exponential Calculus

[Fluffy]

I am currently trying to trace my maths knowledge back down the roots that led to its conception so that I can understand it better. This is largely due to the fact that, at school, we are simply told formulae and definitions with justification or proof and are expected to learn them "parrot fashion". This method of teaching simply doesn't work for me hence the attempt at a differing approach.

We are not told how the formulae for deferentiation and integration of basic functions (ie no chain rule, x^-1 or anything like that) works, we are just expected to learn that if y=x^n, dy/dx=nx^(n-1).

However, I will accept these for now and simply ask for an explanation of the following rule:
y=e^(nx)
dy/dx=ne^(nx)

I understand what the rule is doing and how to calculate it, I just do not know how it was derived.

 

[Ryan2065]

I understand what the rule is doing and how to calculate it, I just do not know how it was derived.

Just a quick note (im studying for finals or the note would be longer) theorems are given in calculus because they are not hard math... The proofs of them, however, are hard math. In my college the calc 1, 2, and 3, classes are in the 100 level math courses... The proofs of many of the theorems in those classes are given in Analysis, which is a 400 level course.

 

[Terrywoodenpic]

I am currently trying to trace my maths knowledge back down the roots that led to its conception so that I can understand it better. This is largely due to the fact that, at school, we are simply told formulae and definitions with justification or proof and are expected to learn them "parrot fashion". This method of teaching simply doesn't work for me hence the attempt at a differing approach.

In the 40's when I was taught calculus we were encouraged not to learn formulae but to think them through, same for all maths, algebra and trig, theorums etc. saved you ever getting stuck in exams.
Remembering back well over 50 years is too much though, and I will have to defer to younger brains.

Terry____________________
Blessed are the pure of heart, they shall behold their God.

 

[Fluffy]

Just a quick note (im studying for finals or the note would be longer) theorems are given in calculus because they are not hard math... The proofs of them, however, are hard math. In my college the calc 1, 2, and 3, classes are in the 100 level math courses... The proofs of many of the theorems in those classes are given in Analysis, which is a 400 level course.

Yeah that is the reason the teachers give for not proving them either. However, keep in mind that the English maths course is much faster than the American one because we do less subjects at the same time when we reach higher levels and I am doing double maths (though I'm not very good at it) so I figure that whilst the formulae proofs are probably ahead of me now, I'm pretty damn close to encountering them (first year of uni probably) and it would help my exams in January so much if I could find them now.

In the 40's when I was taught calculus we were encouraged not to learn formulae but to think them through, same for all maths, algebra and trig, theorums etc. saved you ever getting stuck in exams.
Remembering back well over 50 years is too much though, and I will have to defer to younger brains.

This is just a guess but I would assume that for differentiating, say, you were told that you had to multiply x by the power and reduce the power by one rather than learning the equations I gave in the original post. That is what I prefer to do as well because it makes more sense to me but even this is not quite sufficient because this does not then answer, logically, why the tangent to any given curve should conform to the equation dy/dx=nx^(n-1), why the integral of x^-1 suddenly becomes 1/x etc. Though I am hoping that I can figure out that last example if I can understand why dy/dx=ne^(nx).

 

[michel]

In the 40's when I was taught calculus we were encouraged not to learn formulae but to think them through, same for all maths, algebra and trig, theorums etc. saved you ever getting stuck in exams.
Remembering back well over 50 years is too much though, and I will have to defer to younger brains.

Terry____________________
Blessed are the pure of heart, they shall behold their God.

Same here; frustrating isn't it ? The same goes with me with all the physics equations for DC/AC current..............they're no longer current.