Proof: the derivative of ln(x) is 1/x | Advanced derivatives | AP Calculus AB | Khan Academy

Proving that the derivative of ln(x) is 1/x by using the definition of the derivative as a limit, the properties of logarithms, and the definition of _ as a limit.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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[sciencelover]

I'm always amazed by seeing proofs like this. I feel like I could never think of a proof like this no matter how much time I had.

[elnetini]

I was trying to remember the reason for this result and I finally found the demonstration. Very well explained. Thanks!

[jdlopez]

e=(1+1/r)^r as r goes to infinite. Would you please explain how is it that (1+n)^(1/n) is also e? In all honesty, it checks out - I calculated it in R and it does approx. e. But, mathematically, how are they the same?

[colinreinhardt]

the only thing not 100% clearly explained is moving the limit operation inside of the ln( ) function. Is this always OK? why?

[supersnowva]

This video helped me solve an algorithmic problem! Thank you Sal and Khan Academy! These videos will never get old!

[edbeck]

Very lucid presentation. Thanks. Very satisfying to understand. I used the equation (the one just before you make n substitutions) to prove to myself that the d/dx lnX= 1/X. I plugged a few x values into the n-less equation using for delta X and sure enough the resulting slope = 1/x. To me that was proof enough but I understand the intellectual challenge to show proof without having to plug a single X into the equation. One thing about the rest of your proof presentation (after n substitutions) that I found daunting even discouraging as a beginner calculus student is the seemingly spontaneous and intuitive choice to introduce n= delta X/X. I thought to myself, "man...I'll never get to the level where I can be that intuitive about proofs." But then I remembered recently watching another of your videos where you show that the derivative of e^x is e^x. You start out by defining e as the limit as n>infinity and n>0 with simple equations. Then your subsequent substitution of n into your proof made sense as you mentioned that you wanted to guide your proof to include some semblance of those introductory equations. So I'm encouraged that if I watch enough videos I'll start to build up a repertoire of "mini" proofs to follow even greater proofs.

[sojujinro]

Out of respect and ignorance, lets suppose we didn't know what e^1 was, and we couldn't recognize that the lim as n approaches 0 of (1+n)^1/n was e^1. Then how else could we have proved the derivative of ln x?

[hantuedan]

Thanks for this material... I'm really appreciate it

[xxxxxx-eflj]

4:29 what if x is very close to zero, such as 0.00001?

[zubesR]

awesome video. I just would have loved to see how you could take "limit" from outside to inside of ln.

[killing_gaming]

Ive been confused for a long time and now i understand

[GeoCalifornian]

Thank you for using delta x instead of that silly variable “h” that modern texts use.
/Regards

[stuartyeo]

after a day of thinking I searched up the proof, apparently I wasn't far of. I had done everything up to the n part :(

[ЕвгенияЛысенко-ун]

Why is it that e is (1+n)^(1/n) and not (1+1/n)^n ?

[anna-vbgr]

Thanks for helpful videos you helped me survive my school year last year !!!

[luongmaihunggia]

The derivative of ln(x) is not defined when x is negative but 1/x is defined for when x is negative

[orang]

7:48
See, the only problem I have is that there is a power of 1/0. How can you get rid of that?

[jdlopez]

can you prove why dy/dx of ln(a) = a'/a?