hello and welcome in this video i'm gonna show you how we can use the chain rule to find the nth derivative of some functions find the hundredth derivative of the function y equals e to the negative 2x maybe you think that for finance readings question we have to find the derivative of this function 100 times not really we are not going to find derivative hundreds times we are going to find the first second third or maybe fourth derivative then we can we try to find a pattern in the derivatives and then from the pattern we try to guess the desired derivative which in this case is the hundredth derivative so let's start by finding y prime for finding the derivative of e to the negative 2x we remember that derivative of exponential function is itself remember the derivative of e to the x is e to the x e to the x is the only function that its derivative is itself but here we don't have e to the x we have e to the negative 2x so we have to use the chain rule derivative of e to the negative 2x is itself times by derivative of the inside the inside is negative 2x and the derivative of negative 2x is negative 2. so y prime is negative 2 e to the negative 2x we can put this coefficient in front this is the first derivative now let's find the second derivative we want to find derivative of this function again the negative 2 is a constant and when you want to find the derivative of a function just copy the constant copy the coefficient and take the derivative of the variable part of the function the part that is which variable copy the constant use the constant rule you don't need to use the product rule here so we write the negative two so we have negative two now we have to take the letter of e to the negative two x again but we know that derivative of e to the negative two x is a to the negative two x times negative two if we simplify this negative two times negative two is four i know this but it's better to write it this in this form and in a moment you will understand why i am not going to write it as 4 and i am writing it this form negative 2 to the 2. now let's find the third derivative again we write the coefficient we write the constant first and we take the derivative of e to the negative 2x derivative of that is e to the negative x times negative two again if we multiply this by this negative two times negative two to the two is negative 2 to the power of 3 negative 2 to the 3 times e to the negative 2x if we continue this procedure can you guess that what would be the hundredth derivative attention the first derivative is negative two to the one the second derivative is negative two to the two the third derivative is negative two to the three what is the hundredth derivative if we continue this the hundredth derivative which we want to find would be negative 2 to the 100 and always the original function is here so times e to the negative 2x so this is the hundredth derivative of that function if you want instead of negative 2 to the 100 because the power is even you can simply write 2 to the 100 because when negative raises to the power even negative cancels but you can leave your answer in this form so if you want you can rewrite your answer in the form of 2 to the 100 times e to the negative 2x now look at this question if y is sine of 3x find y prime y double prime y triple prime the fourth derivative and finally we want to find what is the 99th derivative so let us start with y prime derivative of sine is cosine derivative of sine of 3x is cosine theta x but don't forget to multiply by the derivative of the inside and the derivative of 3x is 2. you can write the 3 here but it's always better to write the coefficient in front like this so y prime is 3 cosine 3x now let's see what is y double prime we keep the three derivative of cosine is negative sine multiplied by derivative of the inside which is three three times three is nine and we have a negative but similar to the previous question instead of writing nine it's better to write it as three to the two so what is the second derivative the second derivative is negative 3 to the 2 sine of 3x now let's see what is the third derivative we keep negative three to the two derivative of sine is cosine and derivative of the inside is two three times three to the two is three to that three so the third derivative is negative 3 to the power of 3 cosine 2x what is the fourth derivative it's not bad idea to pause the video right now and try to find the fourth derivative by yourself and then continue watching the video we keep the coefficient derivative of cosine 3x is negative sine 3x times my derivative i'm inside three three times this is three to the four so the fourth derivative is 3 to the 4 sine 3x attention derivative of cosine has a negative and that negative cancels with this negative here if your attention we started with sine of 3x after taking derivative of sine 4 times we got to the original function but we got a 3 to the 4 here so this shows that every 4 time that we take derivative of sine or cosine you can check it later by yourself that for sine and cosine we have this rule every four times that we take the derivative of sine or derivative of cosine we back to the original function look at here derivative of sine is sine again after taking derivative four times of course we have this extra coefficient here and that's because of the inside every time we take derivative inside we get a 3 3 3 and 3. so after 4 times we have 3 to the 4. so far we know that if we take derivative of sine or cosine 4 times we back to the original function so derivative of sine every four time is itself so if we take derivative of sine four times eight times 12 times 16 times every multiple of four its derivative is itself but the 99 is not a factor of four so we cannot conclude that derivative of sine after 99 times is itself but if your attention a number that is close to 99 and is a multiple of 4 is 100 so if we take derivative of this function 100 times its derivative is itself and what we have to put here 3 to the power of 100 y attention for the first derivative we have three to the one for second derivative to the two third derivative three to the three and fourth derivative theory to the four but we want to find the 99th derivative one derivative b for this first of all we don't have 3 to the 100 for the 99 derivative we have 3 today 99. but look at here in the fourth derivative we have sign one step before we have cosine but not only cosine we have a negative also so put the negative here now we know what is the 99th derivative of that function and you can continue backward for finding the previous derivatives for example what is the 99 98 derivative if you want to find for example 98 derivative it is negative 3 to the 98 sine of 3x so it is like a cycle you can go forward and backward now look at this question if y equals 2 to the x find the tenth derivative of that function first we have to find y prime derivative of 2 to the x is 2 to the x l n of 2. let me remind you in general when we have b to the x which b is a number derivative of b to the x is b to the x l and of b so derivative of 2 to the x is 2 to the x ln of 2. what is the second derivative this is a constant this is a coefficient keep it here take derivative of the function you don't need to use any product rule derivative of 2 to the x is again 2 to the x times ln of 2 but we have this n of 2 already here so times ln of 2. which you can write this in the form of 2 to the x times l n of 2 to the 2. can you guess what it would be the third derivative 2 to the x times l n of 2 to the 3. so the first derivative is 2 x ln of 2 to the 1 the second derivative to the 2 and third derivative to the 3 so probably you can guess that what would be the tenth derivative 2 to the x times ln of 2 to the 10. now look at this question find the 100th derivative of the following functions the first function is y equals x times e to the negative x and the second function is sine to the 2x this one is a bit tricky let us start with the first function x times e to the negative x for finding the derivative the first derivative of this function we use the product rule because we have to it's x times e to the negative x derivative of x is 1 times e to the negative x plus we keep x and we take derivative of e to the negative x what is the derivative of e to the negative x is itself times derivative of the exponent which is negative one if we simplify our answer derivative of x e to the negative x is e to the negative x let me simplify it and rewrite it here e to the negative x what is this this is negative x e to the negative x so we can write it in the form of negative x e to the negative x now let's find the second derivative y double prime if we take derivative of e to the negative x it is negative e to the negative x why because derivative of e to the negative x is e to the negative x times derivative of exponent which is negative one it's better to put the negative in front so it is negative e to the negative x now we want to find the derivative of this again we use the product rule derivative of negative x is negative 1 times e to the negative x plus write the first function multiply by derivative of this which is again e to the negative x times negative 1. this negative times this negative is positive and if you add these two together negative e to the negative x minus e to the negative x is minus 2 e to the negative x so let me erase it and rewrite it as negative 2 e to the negative x and here we got positive x e to the negative x so basically we have plus x e to the negative x maybe you can find the third derivative without taking any derivative look at the pattern the first derivative we got e to the negative x and the second one we got negative 2 e to the negative x and in the first derivative we had negative x e to the negative x and this in the second one we have positive x e to the negative x can you guess what would be the third derivative without taking derivative it would be positively e to the negative x minus x e to the negative x if your attention alternatively the sign changes positive negative positive here we had one then two then three and if your attention the sine of x e to the negative x also alternatively changes negative positive then negative if you want to make sure this is right you can take derivative of this and you will see that it would be this now that we have found the pattern we can guess that the hundredth derivative is 100 e to the negative x but not hundred why if your attention in the first derivative the coefficient is positive second derivative is negative so for event derivatives when we take derivative twice four times six times the coefficient of e to the negative x is negative so in the hundredth derivative also we have a negative so we have negative 100 e to the negative x try to find the sine of x e to the negative x attention here is negative in the first derivative and second derivative it is positive third derivative negative so in the hundredth derivative the coefficient of x e to the negative x is positive so we have plus x e to the negative x now let's see how we can find the hundredth derivative for the sine to the 2x y prime derivative of this function is building the parabola two to the four attention sine to the two is basically sine of x to this so now take derivative with the probability the power two to the front subtract the power by 1 so 2 times sine x to the 1 but don't forget to multiply by derivative of inside and derivative of inside is cosine x so the first derivative is 2 sine x cosine x but if you want to find the second derivative from this maybe you can find the answer but it would be a bit harder if you have a good knowledge of trigonometry probably you remember from double angle identities in trigonometry that 2 sine x times cosine x can be written as sine of 2x if you remember we had this rule in trigonometry that the sine of 2 theta is 2 sine theta cosine theta so instead of this instead of 2 sine x cosine x we can write sine of 2 theta this was one of the most important double angle identities if we use this identity so instead of this 2 sine x cosine x we can write y prime as sine of 2x which is simpler why we prefer not to continue from this and we prefer to write this instead of that because if you want to find the derivative of this in this form you have to use the product because you have two sine x times cosine x that's the reason we prefer to use this identity now we have just one term sine of 2x one function we don't need to use any product rule now we want to find that everything about this if you remember from this question that i showed you here every four time that we take derivative of sine or cosine their derivative is itself now back to this question sine to the 2 x we took derivative once how many more times we have to take derivative 99 times because already we have taken derivative once and we have to take derivative 99 times so the question is this what would be derivative of sine of 2x if we take the derivative of this 99 times every 4 times its derivative is itself but 99 is not a multiple of 4 so again we say if we take derivative of this 100 times again 100 times its derivative is sine of 2x times by 2 to the 100 because 100 is a multiple of 4 so derivative of sine is itself but because of the inside derivative of 2x every time is 2 so we have this 2 to the 100 but we want to find derivative of this 99 times not 100. so if you remember from the question i showed you here after four time after four time we got the original function so after 99 times we are we have to move one step back the derivative is cosine with a negative so similar to that if we take the derivative of this 99 times the 100th derivative would be negative 2 to the cosine 2x you should not be confused why i wrote hundred here because already we have a prime here if we take derivative 99 more times it would be the 100th derivative and so on the 100th derivative of this given function here [Music] is this so we took derivative once we got this we use one of the three identities and we rewrote this in this form now we have to find the 99th derivative of this the hundredth derivative of this is itself so 99 is one step b4 then 100 so its derivative is this the reason that we have to put these two to the 99 is because of the inside derivative of 2x is every time 2. it generates a 2 for you at every list i hope you like this video and see you in the next videos