(Proof) The power rule of derivatives (for natural exponents) | Quick Math

Today we'll prove the power rule for derivatives, which will utilize the result of the last Quick Math video.

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[nikodempatrycjuszswiercz]

I know the proof in the video is done like that for obvious reasons, but if anyone's interested, here's how to do it for any n ∈ C (which will prove it for any real n, because R ⊂ C)
we need some tools:
1. d(e^x)/dx = e^x
2. d(ln(x))/dx = 1/x
3. chain rule
4. linearity of the derivative ( d(C * f(x) )/dx = C * d( f(x) )/dx, where C is a constant)
5. properties of logarithms, exponentiation and multiplication/division (which i will not emphasize the usage of, as they're quite trivial)
for the proof of which we don't need the power rule
let's start with what we want to calculate and rewrite it
d(x^n)/dx = d( (e^ln(x))^n )/dx = d( e^(n * ln(x)) )/dx
which we can use together with our "tools" to simplify, first using 1 and 3
= e^(n * ln(x)) * d( n * ln(x) )/dx
then use 4 to get "n" out of the uncomfortable place and after that use 2 to finish taking derivatives
= (e^ln(x))^n * n * d( ln(x) ) = x^n * n * 1/x
now we just need to tidy up this expression to get the form we want
= n * x^n / x = n * x^(n-1)
that is exactly the result we wanted to get
now, just to clarify the situation with x = 0,
for n < 0, the function is not continuous (nor is it even defined) at the point x = 0
for n = 0, it's a constant function, so the derivative is equal to 0
for 1 > n > 0, the slope of the tangent line to the function's graph at x = 0 is "vertical", making the slope undefined (i think? i don't really know whether it's undefined, undeterminate or anthing else, but the sure thing is, it doesn't work)
for n = 1, the derivative is going to be 1, disregarding the argument
for n > 1, the derivative is 0, which can be derived from the resulting formula
and i think it's a good place to stop, if there's any mistake, please do comment back : )

[AndrewDotsonvideos]

Prove that for andrew in the basement there exists Flammable maths stuck in the closet?

[EpicMathTime]

Sorry for the late upload (this was intended for last Friday). I keep forgetting that After Effects has a render engine built during the Spanish inquisition.

[captainsnake]

"by definition, this is equal to this that and the other thing"

me when writing a proof

[jetblack]

I'd really like to see a proof on the topological space homeomorphic to the 2-torus which allows Andrew Dotson and Pappa Flammy to simultaneously exist in each other's basement.

[alkankondo]

Cool! Now I won't feel "guilty" using some arbitrary "power rule" formula; I now know why it's true!

[WrathofMath]

As slick as an oiled up salmon! Beautiful!

[brunojambeiro]

Please do a prof of why there isn’t a quintic formula. I haven’t seen any video on youtube explain it properly. It won’t be a quick video but i would still watch it anyway.

[harrypotter]

There’s a nice proof for rational exponents which invokes the binomial theorem in a similar way.

[martinwilke]

You were so quick I had to pause a number of times to really follow you.

[waywardson]

My calc series professors actually “proved” these for us in a non-rigorous way.

[lorisdeplano]

I was listening to The Pot just before I watched the video, great choice!

[maureendotson]

Great video! I don’t understand anything like I don’t understand on Andrew Dotson’s videos. 😂. ~ Andrew’s Mom 💕

[justacutepotato]

Your voice is beautiful. And so is the background. :)

[Smitology]

Can't you do it using induction, assuming the derivative of a constant and the product rule (both of which can be proven from first principles)?

[subscribetopewdiepie]

I know you are busy but you should upload more regularly, your content is amazing.

[jagula]

Can you make a video on how you write like this? Great work and gl on 20k

[brooksbryant]

Could you do a video explaining how an open sets are related to donuts and spheres? I’ve never understood how you get from one to the other

[Felixkeeg]

I'm glad that my puny Chem grad student brain can follow along these proofs

[NikolajKuntner]

Given the style of presentation, I'd like to see some proofs with pictures included.