how do we know the derivative of ln(x) is 1/x (the definition & implicit differentiation)

We will show that the derivative of ln(x), namely the natural logarithmic function, is 1/x. We will use the definition of the derivative and also implicit differentiation.
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[wildramen]

I have graduated 3 months ago, at the start of the calculus class 2 years ago i hated calculus but here i am, loving calculus and enjoying every second of your awesome videos.

[idomins]

You know its about to get real when he starts using the blue pen (-:

[soyalguienyt]

10:43 I really love the satisfaction I get when my mind snaps and know how the demostration continues before the video. Great video!

[jimbeasley]

How do we know ln(x) is a logarithm? I once had a professor “define” ln(x) as a function whose derivative is 1/x. He then proceeded to show the ln(x) is indeed a logarithm, and it has the base e. I’d like to see this again. It was very inspiring, but I have forgotten how it was done.

[whozz]

Another proof using parametric equation:
x = e^t
y = t

dx/dt = e^t
dy/dt = 1

(dy/dt)/(dx/dt) = dy/dx = 1/e^t = 1/x

[magellan]

Just watched it again as there were a few things I wasn’t sure of. I really liked it when he explained one trick to use was because the natural log is a continuous function, and the limit of a continuous function is a continuous function of the limit, you can move the limit inside the parentheses to simplify things. Cool stuff.

[ezvac]

Another way :
exp(ln(x)) = x
Derivative of both sides :
ln(x)' * exp(ln(x)) = 1
Replace exp(ln(x)) by x and divide the whole equation by it :
ln(x)' = 1/x

[marcushendriksen]

Great video man! I feel like you've made me so much smarter; this time I was actually able to see ahead a little bit, that the argument of ln would end up being e^1/x (this was around when you brought the derivative into the u world)

[sajidrafique]

I loved calculus in college and now i am 71 and watch these videos to fill the gaps in my understanding and keep dementia away

[itamarbentwich]

I really enjoyed your last few videos, and I am glad you're back to uploading more videos containing your explanations

[magellan]

I have always wanted a more detailed explanation of this result. This is the best I’ve seen on the subject. Considering things like Euler’s identity and the quantum wave equation and other uses of the exponential function, it seems to me it’s the most useful of all the special functions.

[rajendramisir]

Elegant proofs for the derivative of ln(x). I like the intelligent and creative ways you used to develop and establish your proofs. Thanks.

[iansheridan]

You are my new favorite high school math teacher. In my AP calculus class, we were never taught how to derive this. Only taught to memorize that d/dx ln(x) = 1/x

[hopp]

blackpenredpen could you solve the non elementary integral of x^x. You did the (easier) derivative so please do the difficult integral or let Payem do it

[vegasuser]

Wonderful videos. It is a long time ago that I studied complex variables, differential and integral calculus and algebra. So it is great fun watching this guy do with ease what most of us struggled with when learning the basic elements of these important mathematical techniques. I can generally follow him right to the end once I see where he headed. The mathematical manipulations seem to be firmly rooted in my brain. The algorithms he applies for problem solving are much less so.

[FF-pvht]

Its a shame we dont get teached this stuff in school but are just supposed to remember f'(x)=1/x of F(x)=ln(x)

[SudarshanBaurai]

Dear friend, you are not only genius but you a great guru (teacher). My regards - Sudarshan🙏

[gebcrafter]

this math professor dripping out with tha preme jacket

[jakehu]

I have a fourth proof: If we differentiate e^ln x, instead of resulting in x, we use the chen lu, where u = ln(x). That results in e^(ln x) * du/dx. However, if we use the power rule, it results in 1. Therefore, x * du/dx = 1. We solve for du/dx = 1/x.

[AkiyamaKatsuko]

I've always been told that the derivative rule for f'(x) of ln(x) has always been 1/x but I've never understood how that was proven. Thank you for the explanation.