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# Proof: the derivative of ln(x) is 1/x

## Video transcript

what we're going to do in this video is prove to ourselves that the derivative with respect to X of natural log of X is indeed equal to 1 over X so let's get started so just using the definition of a derivative if I were to say the derivative with respect to X of natural log of X that is going to be the limit as Delta X approaches 0 of the natural log of X plus Delta X minus the natural log of X all of that over Delta X so now we can use a few logarithm properties we know that if I have the natural log of a minus the natural log of B this is equal to the natural log of a over B so we can use that right over here where have the natural log of something minus the natural log of something else so all of this is going to be equal to the limit as Delta X approaches 0 of the natural log of this divided by that so X plus Delta X over X X plus Delta X over X all of that over Delta X and actually let me just write it that way all of that over Delta X once again all I did if I have natural log of that minus natural log of that that's the same thing as natural log of this first expression divided by that second expression come straight out of our logarithm properties well inside of this logarithm X divided by X is 1 and then Delta X divided by X we could just write that as Delta x over X so that's another way of writing that and then we could put a 1 over Delta X out front so we could say this is the same thing as this is equal to the limit as Delta X approaches 0 of I'll do this in another color so this I can rewrite as 1 over Delta x times the natural log of one plus one plus Delta X over X let me close that parentheses so now we can use another exponent property if I have I'll write it out here if I have a times the natural log of B that is equivalent to the natural log of B to the a and so here this would be the a in that case so I could bring that out and make that an exponent on this so this is all going to be equal to the limit as Delta X approaches 0 of the natural log of give myself some space 1 1 plus Delta X over X to the 1 over Delta X power 1 over Delta X power now this might start to look familiar to you it might start to look close to the definition of e and we are indeed getting close to that in order to get there fully I'm going to do a change of variable I am going to say let's let let's let n equal Delta X over X Delta X over X well in that case then if you multiply both sides by X you get Delta X is equal to n X again just multiply both sides of this equation by X and swapped the sides there and then if you wanted 1 over Delta X 1 over Delta X that would be equal to 1 over N X which we could also write as 1 over N times 1 over X actually let me write it this way actually that's the way I do want to write it so these are all the substitutions that I want to do my change of variable and we also want to say well look as Delta X is approaching 0 what is n going to approach well as Delta X approaches 0 we have n will approach 0 as well 0 over X well that's just going to be 0 for any X that is not equal to zero and that's okay because zero is not even in the domain of natural log of X so this is going to be for our domain it works that is Delta X approaches zero and approaches zero and you can think about the other way around as n approaches zero Delta X approaches zero so now let's do our change of variable so if we make the substitutions instead of taking the limit as Delta X approaches zero we are now going to take the limit as n approaches zero of the natural log of myself some parentheses and I'll say one plus and now this is the same thing as n 1 plus n and then all of that is going to be raised to the 1 over N times 1 over X that's what 1 over Delta X is equal to this is 1 over Delta X right over here which we have over here and it's the same thing as 1 over N times 1 over X so let me write that down so this is the same thing as 1 over N times 1 over X now we can use this same exponent property to go the other way around well actually let me just rewrite this another time so this is going to be the same thing as the limit as n approaches 0 of the natural log of 1 plus I'll just write this in orange 1 plus n to the 1 over n if I raise something to exponent and that's times something else I can that's the same thing as raising it to the first exponent and then raising that to the second value this comes once again straight out of our exponent properties and now we can use this property the other way to bring this one over X out front but in fact the 1 over X itself is not affected as n approaches 0 so we can even take it completely out of the limit so we could take it all the way over there and this is when you should be getting excited this will be equal to 1 over x times the limit and approaches zero of the natural log of 1 1 plus n we do that orange color 1 plus n to the 1 over N and now then what really gets affected is what's going on inside of the natural log that's where all of the ends are and so let's bring the limit inside of that so this is all going to be equal to get myself some space a little bit of extra chalkboard space this is going to be equal to 1 over x times the natural log times the natural log of the limit as n approaches 0 of 1 plus n to the 1 over N close those parentheses now this is exciting what is inside the natural log here well this business right over here this is a definition of the number e so that is equal to e well what's the natural log of e well that's just 1 so it's 1 over x times 1 well that is indeed equal to 1 over X which is exactly the result that we were looking for that the derivative with respect to X of natural log of X is 1 over X very exciting