Question

Did he ever explain why the derivative of powers of x always has a footnote limiting it to integer powers?

Or is there a proof i could refer to that demonstrates why the restriction is necessary?

*edited* - powers of n --> powers of x

Answer 1

are you asking about the power rule
\[ \frac{d}{dx} x^n = n x^{n-1} \]
?
See https://en.wikipedia.org/wiki/Power_rule
notice that the power rule generalizes to any n e.g. irrational numbers.

Answer 2

Yes! Sorry i should have said powers of x, not powers of n. Thank you for the reference.

So if it generalizes, does anyone know why it's written in the lectures with the additional list of integers?

Answer 3

I think they proved it using the binomial theorem
https://en.wikipedia.org/wiki/Binomial_theorem#Geometric_explanation
http://www.wyzant.com/resources/lessons/math/calculus/derivative_proofs/power_rule
which is ok for integer powers. They did not want to claim more than was actually proven.

Answer 4

That is indeed how he demonstrated it originally. Thank you very much for clearing it up.

Answer 5

Positive integers are easy to calculate based on difference quotient. Then after deriving the division rule of derivatives negative integers are easy to calculate as reciprocals.Then implicite differentiation can be used to the equation y^m=x^n where m and n are integers to get the rational potenses and finaly after e^x is differentiated the general rule how to differentiate x^r for all reals can be proven.

That's long road but it's based on telling to the stundent only things that they can derive themselves using the tools they've been taught.

This is btw. the same sequence that is used to teach the numbers. First we start with postive integers for counting sheeps, then we sell the sheeps for credit and need negative numbers for debit, then we need to divide the lumber for the expanded fence and we get rational numbers for division of whole numbers. When we start to figure out the diagonal of the field and we need square root of 2 and we figure out algebraic reals and finally things like pi and e give us transcedential reals. You cannmot get evrything at once.

Answer 6

@topi
I think the lecturer perhaps assumes more knowledge than i have retained from many years ago or maybe is just unaware that this kind of contextual information is not obvious.

Getting everything at once is not really the issue i think - just knowing what i have and do not have is the first step and you guys explained that. Thank you very much.