 Hi, I'm Zor, welcome to Unizor Education. I would like to continue exemplifying derivatives for certain basic functions and today's function which we will consider is logarithm. Well, this lecture is part of the course of advanced mathematics for teenagers. It's presented on Unizor.com. I suggest you to watch this lecture from this website because every lecture contains on this website contains some notes which are very useful like a textbook basically plus registered students can take exams for instance and participate in other purely procedural educational steps. Alright, so derivative of the logarithm. Well, first of all I would like to consider natural logarithm. So you remember that function natural logarithm of X is basically logarithm based with a number E which is approximately 2.71 but this is the fundamental constant of calculus. We talked about this number E in particular as if you remember if you have some kind of an exponential function a to the power of X then whenever this particular tangential line at X is equal to 0 is at 45 degrees if a is equal to some number E 2.71 blah blah it's irrational number. Another there was an amazing climate which we were considering before 1 plus X to the power 1 over X goes to number E as X tends to 0 that's another definition of E they're all equivalent to each other. Well, remember this one actually we will be using it in this lecture. So let me just wipe out this picture and go to actually I put it somewhere else and go to the calculation of the the calculation of the derivative of natural logarithm of X. So let's just go to definition right what's the definition of the derivative it's function increment divided by argument of increment of the argument as limit of delta X as increment goes to 0 at any particular point X. X is a fixed number which is part of the domain for a function f of X. So in this particular case we are talking about limit of delta X goes to 0 of logarithm X plus delta X minus logarithm natural logarithm of X well I don't really need this bracket I can put it here divided by delta X right. So this is basically a definition where X belongs to the domain of the logarithm which is positive numbers obviously and delta X is some increment which is an infinitesimal variable. X is fixed delta X is infinitesimal variable. Well now to calculate this we have to like make some transformations right obvious transformation. You remember that the difference between logarithms of the same base is basically logarithm of the ratio right remember this. Logarithm v let's say u divided by v is equal to logarithm u minus logarithm v well this is just a elementary formula of logarithm right. So the difference between logarithms we can replace with logarithm of their ratio divided by delta X right. Well delta X tends to 0 alright equals. Now I can obviously divide this to get logarithm of 1 plus delta X divided by X and delta X right that's the same thing divided X by X is 1 delta X is delta X over X. Now here is an interesting story let's do it this way I will divide by X and multiply by X. X is a positive number so basically nothing is changed here and now consider this and this. Well X as I was saying before is a fixed number within the domain of the logarithm and delta X is infinitesimal so basically delta X divided by X is just an infinitesimal and it goes to 0 as well so it's logarithm of 1 plus I will use delta divided by delta times X right where well I have to put the limit here delta goes to 0 right. Now let's go back to our amazing limit remember this now if this goes to this then if I will apply logarithm to both sides logarithm is a continuous function so basically if this goes to this then logarithm of this goes to logarithm of this because logarithm is continuous function and what does it mean well let me put it here natural logarithm of 1 plus X to the power 1 X goes to natural logarithm of E right as X goes to 0. Now this is an exponent and I know how logarithm is working with exponent you have to factor it out so it's 1 over X logarithm 1 plus X. Now what is natural logarithm of E well obviously it's 1 because this is the power which you have to raise the base which is E to get E which is 1 so we have this particular limit as an immediate consequence of this amazing limit which I was discussing in one of the previous lectures right now what's the difference between this and this well there's only one difference well this X is infinitesimal right so don't confuse this X with this X I was just using the letter X in my amazing limits thing so that's why I continue I can as well put delta here this is a variable it's not a constant so let me put delta here and it would be more looking more like whatever I have here X is basically a constant and delta is infinitesimal and delta is infinitesimal here so the difference is only this and this is the constant which means I have actually I have to put it outside of the limit since it's a constant so what's remaining while remaining this logarithm 1 plus data divided by delta as delta goes to 0 which is 1 so there is only thing which is remaining is 1 over X and this is the derivative of natural logarithm so f derivative of X in this case is equal to 1 over X but just before I go any further let me tell you that personally I felt it a little bit strange this particular result because this is X to the power of something in this case it's X to the power of minus one now we know that X to the power of n derivative is equal to n times X to the power of n minus 1 right and this is not only for n integer actually it's for any n and including negative by the way so that's why I was kind of surprised to see that X to the power of minus one is the result of this particular derivative because it looks like it belongs to a completely different class of functions the class of functions which which have something in in the power like X to the power of something so this looks like one of these but at the same time well again that's the basically the surprise which I had I did not expect to have this particular result for for logarithm well that's what it is I mean I can't say anything more than that it's a little bit surprising the first time when I saw it but that's basically it all right so let me just continue this and go to logarithm with any other base so let's say you have g of X equals to log of X with a base B which is any positive number B as a base so what is the derivative well it's actually easy because logarithm X is equal to logarithm X with a base any other base so there is a formula basically which transforms logarithm by with with one base to logarithm with another base and this formula was in detailed considered when I was talking about logarithms I do suggest if you don't remember it to refer to that particular lecture so and this is natural logarithm X divided by natural logarithm B which means that is equal to logarithm X divided by logarithm B derivative and this is a constant obviously so this is 1 over X times logarithm B so derivative of natural logarithm of X is 1 over X and this is just a constant multiplier I retain it in the denominator well that's it it's a short lecture which is kind of very easy but I was just trying to exemplify a particular function so we were considering different examples we were considering power function exponential function logarithmic function there is another lecture which is trigonometry trigonometric functions so these are basic functions from which well any kind of a algebraic or calculus problems actually consist of and to take the derivative you just probably have to combine different properties of the derivatives which we are addressing in a different lecture with derivatives of elementary functions which I have exemplified in these few lectures well that's it for today thank you very much and good luck