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Current time:0:00Total duration:7:03

Proof of power rule for positive integer powers

Video transcript

i just did a uh several videos on the binomial theorem so i think it's and now that they're done i think now is a good time to do the proof of the derivative of the general form so let's take the derivative of x to the n and now that we know the binomial theorem we have the tools to do it so how do we how do we take the derivative well what's the classic definition of the derivative it is the limit the limit as delta x approaches zero of f of x plus delta x right so f of x plus delta x in this situation is x plus delta x to the nth power right minus f of x well f of x here is just x to the n all of that over delta x and now that we know the binomial theorem we can figure out what this what the expansion of x plus delta x is to the nth power and if you don't know the binomial theorem go to my precalculus playlist and watch the videos on the binomial theorem so the binomial theorem tells us that this is equal to and let me i'm going to need some space for this one the limit as delta x approaches zero and what's the binomial theorem channels this this is going to be equal to i'm just going to do the numerator x x to the n plus n choose one and once again review the binomial theorem if this is looks like latin to you and you don't know latin n choose 1 of x to the n minus 1 delta x plus n choose 2 x to the n minus 2 that's x n minus 2 delta x squared right delta x squared and then plus and we have a bunch of the digits and in this proof that actually you know we don't have to go through all the digits but the binomial theorem tells us what they are and of course the last digit you know we just keep adding is going to be 1 well you know you could it would be n choose n which is 1. let me just write that down n choose n it's going to be x to the 0 x to the 0 times delta x times delta x to the n right so that's the binomial expansion and let me switch back to minus so all of that's what i that green that's x plus delta x to the n so minus x to the n minus x to the n power that's x to the n i know i squanched it there all of that over delta x let's see if we can simplify so first of all we have an x to the n here and at the very end we subtract out an x to the n so these two cancel out right and then if we look at every term here every term in the numerator has a delta x right so we can divide the numerator and the denominator essentially by delta x right we could this is the same thing as 1 over delta x times this whole thing so that is equal to let me the limit as delta x approaches 0 of so we divide the top and the bottom by delta x or we multiply the numerator times 1 over delta x we get n choose 1 x to the n minus 1. what's delta x divided by delta x well that's just 1 right plus n choose 2 x to the n minus 2. this is delta x squared when we divide by delta x we just get a delta x here delta x and then we have you know we keep having a bunch of terms we're going to divide all of them by delta x right and then the last term is delta x to the n but now we're going to divide that by delta x so the last term is becomes n choose n and choose n x to the 0 is 1 so we can ignore that delta x to the n divided by delta x well that's delta x to the n minus 1. right and then what are we doing now well remember we're taking the limit as delta x approaches zero so as delta x approaches zero pretty much every term that has a delta x in it it becomes zero right we just you know when you multiply by zero you get zero so every term after this first term has no delta x in it but every other term does every other term even after we divided by delta x has a delta x in it so that's a zero every term is zero all of the other i don't know what is it n minus one terms they're all zero so all we're left with is that this is equal to n choose one of x to the n minus one and what's n my n choose one that equals n factorial over one factorial divided by n minus 1 factorial times x to the n minus 1. 1 factorial is 1. and you know if i have 7 factorial divided by 6 factorial that's just 7 or if i have 3 factorial divided by 2 factorial that's just 3. you can work it out 10 factorial divided by 9 factorial that's 10. so n factorial divided by n minus 1 factorial that's just equal to n so this is equal to n times x to the n minus 1. so that's the derivative of x to the n n times x to the n minus 1. so we just proved the derivative for any well for any positive integer when the with x to the power n where n is any positive integer and we see later it actually works for all uh actually real numbers in the exponent um so i will see you in a future video see ya oh and another thing i wanted to point out is you know i i said that we had to know the binomial theorem but if you think about it we really didn't even have to know the binomial theorem because we knew you know in any binomial expansion i mean you'd have to know a little bit but if you did a little experimentation you would realize that whenever you expand you know a plus b to the nth power the first term is going to be a to the n and then the second term is going to be plus n a to the n minus 1 b right and then you could keep having a bunch of terms but these are the only term terms that are relevant in this proof because all the other terms get cancelled out by you know when delta x approaches 0. so if you just knew that you could have done this but it's much better to do with the binomial theorem ignore what i just said if it confused you i'm just saying that we didn't you we could have just said oh there you know there's the rest of these terms just all go to zero anyway hopefully you found that fulfilling i will see you in future videos