Derivative of 1/x without the power rule

[gaunt kite]

I'm following 3Blue1Brown's Essence of Calculus series, and near the end of Chapter 3; he proposes the following question:

Imagine a rectangle in 2 dimensions, whose area is 1. Let's say that its width is x; the height has to be 1/x to maintain that area of 1. 

For the derivative, imagine nudging up that value of x by some tiny amount, dx. How must the height of this rectangle change so that the area of the rectangle stays as 1? That is, increasing the width by dx adds some new area to the "right". So, the rectangle must decrease in height by some d(1/x); so the area lost off that top cancels out the area gained. You should think of that d(1/x) as some negative amount, since it's decreasing the height of the rectangle.

Now, try reasoning out the slope ( d(1/x) / dx ) is. Compare what you got instead of what you would have gotten if you blindly applied the power rule ( nx^n-1)

I've attached a visualisation of this problem and my attempt at solving it. I've only gotten as far as x+dx^2 - x*dx.
(Sorry for the sloppy lines, don't have any graph paper left)

[craggy hornet]

why even do it without the power rule in the first place?

wut is the objective behind this?

[gaunt kite]

for a better understanding of derivatives and finding the slope?

you dont really learn alot if you just slap on the power rule

[craggy hornet]

you're new to this topic?

[gaunt kite]

yes

[craggy hornet]

oh alr

okay. look at the step you did after
$(\frac{1}{x + dx} - \frac{1}{x})\frac{1}{dx}$

[mental mesa]

Galactic Fur

[craggy hornet]

you have made the first term 1/(x + dx^2)
meaning, you only multiplied it to dx only, not to x

and also. could you tell me how you got the exponent -1 on each term there?

[gaunt kite]

1/x+dx * 1/dx

doesnt that equal 1/x+dx*dx which i rewrote as 1/x+dx^2

[gaunt kite]

well (1/n)^-1 = n ; did that so i could get rid of the fraction

[cerulean raptor]

Ok I see what happened

You are trying to prove the derivative of 1/x using the limit definition

2 things in your work: Your 1/dx should be distributed to 1/x+dx as 1/x*dx+dx^2

I think this is what Galctic Fur was trying to get at

And, on the last step, if you wanted to put -1 exponents, you would have to do it to both fractions together, and I don’t know why you would do that

Here’s what I would do

Sorry I don’t know how to use texit

[gaunt kite]

what exactly did you do in the 3rd step?

[orchid wyvern]

Hi! It appears he multiplied the subtracting fraction by (x+dx)/(x+dx), so that it could share similar terms. However, he also forgot to multiply X to the first fraction, creating an x^2-x-dx. either way, negative dx is scary and shouldn't be worked with unless you really know what you're doing.

[winged plover]

Ah, I watched that video and there were two such examples. I believe he has an explanation on there for one of them? And for the second one, I think that you can find satisfactory explanations in the comment section. That's what I did

And I dont recommend watching essence of calculus if you're new to calculus. It starts to get really complicated and weird after the 4th episode. You should instead watch Khan academy's videos on the same, I think he'll be able to explain the concept better.

[cerulean raptor]

Yea I forgot to write it, but I remembered to add it into the combined fraction

[cerulean raptor]

Also no it wouldn’t create an x^2-x-dx, i just forgot to put an x on the bottom

[gaunt kite]

i dont really like Khan Academy's explanation of calculus, feels like it's just preparing me for a class instead of actually teaching me calculus

[gaunt kite]

there were a couple people who posted viable solutions in the comments, but i didnt really understand them

[winged plover]

I found them helpful but you may not. It's best to stick with whatever one likes most

[winged plover]

I see

[gaunt kite]

mostly because youtube has a horrible formatting and you cant even do a new line proplery

[gaunt kite]

why are there limits? i was able to find the derivative of x^2, x^3, etc. etc. without using the power rule and without limits

[mental mesa]

OccamsChainsaw

[cerulean raptor]

Its the same thing

saying dx^2 is negligable is the same as saying dx goes to 0

[cerulean raptor]

What I'm doing is the formal definition of a derivative

[gaunt kite]

ah oke, i dont really care/mind about using any set definitions, im more or less struggling with actually solving the "inner part" of the equation which is

[mental mesa]

OccamsChainsaw

[gaunt kite]

(this is what i get for trying to copy paste from overleaf)

[cerulean raptor]

yea so what I did was combine the numerator into one fraction

[gaunt kite]

but since they have unlike denominators, i thought you couldnt. i obv. know you can multiply by (something) to get the denominators to be like but i couldnt figure out what to multiply the one of the right (1/x) by

[cerulean raptor]

But you can make them have the same denominators

[gaunt kite]

what should i multiply them by to have them have the same denominator?

[cerulean raptor]

You multiple each of them by the other fraction

So you'd multiple 1/(x+dx) by x/x

and 1/x by (x+dx)/(x+dx)

idk how to use texit

[mental mesa]

Neemi Hata
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[gaunt kite]

OHHH

that makes

sense

[mental mesa]

OccamsChainsaw

[gaunt kite]

like this?

[cerulean raptor]

Ya

Well its actually the other way

like the right fraction should be on the left

[gaunt kite]

why multiply/"distribute" (1/x+dx) to the (1/x) fraction instead of just (x+dx)