[Music] Hey friends, I hope you're all very well, welcome to the derivatives course. Now we'll see how to find the derivative of a quotient or of a division. In this video, we're going to find the derivative of this function, which, as you can see, is the derivative of a division or of a quotient. So, first of all, the little formula, which, well, it 's not very easy to understand, but we'll see that it's simple. If there's a function that is the division of two functions, look, above would be gdx and below hdx. To identify them, I'm going to put them above, the one above is called g of x and the one below is called hdx. It doesn't necessarily have to be the xy axis or dhx, but by identifying them, no. So, how do you find the derivative? The derivative is as follows: the one below and the one above, we always start with the one below, the one below times the derivative of the one above minus the one above times the derivative of the one below divided by the one below squared. This may seem very difficult, but with this exercise and the one that I'm going to leave you for practice, I hope it will seem a little more easy obviously this is a very simple exercise after we see in a video later on we're going to see something called the chain rule we're going to continue seeing more examples of derivative of a quotient in case you want to look for it so the first thing I recommend is to find those derivatives yes here I'm going to place the derivative above and here the derivative below yes you will see how much simpler it is not the derivative above then the derivative of 2x is 2 - the derivative of s of 3 which is zero or the derivative above is 2 now the derivative below and we lower the exponent we place the x and we subtract 12 minus 11 all of this we already saw in the previous videos if you until now this is the first video that you see I invite you to watch the previous ones and well later on we are also going to do many more exercises now yes we are going to start to find the derivative of the function so here I write the derivative of the function f x is equal and we are going to do a division and here we are going to see the service of this that we did here no then it is I do not like learn them and if not simply the one below by the derivative of the one above minus the one above by the derivative of the one below and now if I tell you again the one below which is x squared by the derivative of the one above but the derivative of the one above we already did is y minus and we change the one above which is 2 x 3 generally that is placed in parentheses the one above by the derivative of the one below which is not this one but this one that we already did here by 2 x we always divide by the one below squared then that x squared we raise it to the square and there we could say that the division is finished but in mathematics remember that you always have to do all the operations that can be done so we are going to do all the operations then I continue writing over here the derivative of the function fx is since we multiply x squared by 2 well generally it is organized and it becomes 2 x squared batteries always in the division batteries with this negative because this negative is going to affect everything that comes after then I am going to do this operation which is a binomial multiplied by a monkey of mine that Remember that the way to do it is my monkey is multiplied by the two terms of the binomial but remember that all of this is affected by the negative so after that negative what you do is place a parenthesis then - and we place a parenthesis where we are going to write this whole operation and then 2x by 2 x 2 by 2 4 and x x x x squared remember that when we multiply letters what we do is add the exponents here it says x to the 1 and x to the 1 so one plus 12 now minus 3 times 2 x to the 1 so 3 times 2 6 and since there is only one x that x remains above and below so we do the operation not x squared squared remember that when something has two exponents others are multiplied by 24 x to the 4 now what do we do now well here you can remove the parenthesis so this negative goes for the two terms of the parenthesis I'm going to continue here at the top then I place efe derivative of x is equal and here it says 2x squared and the negative for both then this negative what it does is multiply by the signs inside then minus x change the signs inside no here it is plus remains minus 4x squared and here this remains plus then 4x squared plus 6x we can now remove the parenthesis above and below we continue writing x to the 4 we continue with the operations look that up here there are similar terms because there are two terms that have x squared then I write the derivative of fx it is equal and we do that subtraction of similar terms no then 2 x squared minus 4 x squared 2 minus 4 which is minus 2 and we are adding and subtracting x squared plus 6x below it says x to the 4 as you observed that all the terms have the x then this is not very normal but here you can factor above the x then I am going to do it here it would be efe of x equals and factor made the x above x factor of yes because factor made well the number there is no need because I am going to eliminate it with the 4 because factor made the x because look that it is repeated in the two terms then x factor of here it would be minus 2 x remember that in the factor what we do is place the result of dividing this by the x then yes / minus 2 x squared by x is minus 2 x plus and if divided by 6 x gives 6 over and below we write x to the 4 the truth well that's all suddenly not even I should continue doing it because well the idea of the video was that we will practice with the derivative of one with a quotient so we already practiced that but well the idea was to finish the exercise yes because well you are going to practice this a lot here is x which is already as a factor it is eliminated with one of the four in the back then the derivative of fx remains it is the same we eliminate this x with one of these four non-chords that here there are four and below there will be 3 no then it would be I can now remove the parenthesis minus 12 x plus 6 over x cubed because one of the ones below is eliminated and here our derivative ends as always finally I am going to leave you an exercise for you to practice you already know that you can pause the video you are going to find the derivative of this function which again is a quotient obviously and the answer will appear in 3 1 first of all well the recommendation that I give you is to take the derivative from above and from below not the one above would be gdx and the one below hdx the derivative of the one above would be 3 times 2 6 x and the derivative of 5 which is 0 the derivative of the work 3 times 2 6 and the exponent is subtracted from 1 so it is no longer three but two now the one below times the derivative of the one of rivas 6x minus the one above this binomial times the derivative of the one below that 6 x squared over the one below squared if we multiply 2 by 6 12 x to the 3rd power x x to the 1st power gives x to the 4th - and we open parentheses batteries with this not the monkey the binomial by the monkey thousand 3 times 6 18 and x squared times x squared x to the 4th plus 5 times 6 30 x squared here let us remember that that square goes to the 2 and to the x to the cube no or rather it would be 2 squared which is 4 and x cubed squared gives x room here the negative we place it at the two that is why this one that was positive becomes negative and this one that was positive also becomes negative these two similar terms we add or subtract 12 minus 18 minus 6 x to the 4 and here simply as I see that it is going to be able to simplify then factor did 2 x squared here I clarify something here it could have been factored 6 even negative minus 6 x squared but I did not factor I did the 6 because well it would not have been possible to eliminate let's see here factor I did 2 x squared and minus 6 x to the fourth divided by 2 x squared minus 3 x squared and minus 30 x squared divided by 2 x squared of 15 nothing more and for what does this say here the 2 we can say batteries that here this 4x cannot be simplified with the ones above yes because there is a subtraction it is not always simplified it is when there is a factor or well there is a way to simplify but it is that there are no teachers generally we explain it by what so that they do not get confused it is simply simplified but when there is a factor outside if here for example half of 21 and half of 42 and this x squared we eliminate it with two of these x's then below would be x to the 4 if batteries that here can no longer be eliminated because there is a subtraction then what we have left this has already been eliminated all we have left is minus 3x squared minus 15 and below would be 2x to the 4 well friends I hope you liked the class remember that you can see the complete derivatives course available on my channel or in the link that is in the description of the video or in the card that I leave here at the top I invite you to subscribe comment share and give the video and I do not feel anymore bye bye