Wednesday, February 22, 2012

An interesting pattern with derivatives

I found an interesting pattern while finding  higher order derivatives using the power rule. For readers who are not familiar with derivatives you might want  to watch the video here ( MIT open course ware derivatives.)

I was curious what the 3rd derivative of  x5 was.
So I used the power rule  to find the first derivative and I got f1(x)= 5x4 I applied the rule again to get the second derivative and got f2(x)= 20x3 I once again used the power rule for the third derivative and found the third derivative was f3(x) = 60x2 Now by this time I started to get curious as to what the Nth derivative,fn(x) would be for functions using the power rule. So I began to to work it out algebraically. I started with the general function f(x)= xa and began taking the derivative by continuously using the power rule, and got the following results:

  1. 1(x)= a x a-1
  2. 2(x)= a (a-1) x a-2
  3. 3(x)= a (a-1)(a-2) x a-3
  4. 4(x)= a (a-1)(a-2)(a-3) x a-4
  5. 5(x)= a (a-1)(a-2)(a-3)(a-4) x a-5
From these a pattern begins to emerge with, a , as I took the higher order derivatives. Here is the pattern of  ,a, throughout each of the derivatives:
  1. a
  2. a (a-1)
  3. a (a-1)(a-2)
  4. a (a-1)(a-2)(a-3)
  5. a (a-1)(a-2)(a-3)(a-4)
As we can see it looks like a pattern similar to a! or a factorial, but this alone is not the pattern we need to a be able to see how this relates to the nth derivative. Well lets look at the derivatives. The second derivative of  f(x)= xa  is f2(x)= a (a-1) x a-2 ,and from this we see that it is not just a! it is a!/(a-2)! because a! keeps going until (a-some number) = 1  For the second derivative to make sure a! does not go past a (a-1) we use (a-2)! in the denominator because it cancels out all of the other factors past that point.  That brings us to the general form of a!/(a-n)!

Now with the a in the exponent  xwe see that the a reduces by the the derivative we are on for example the second derivative ,  f2(x), the exponent is a-2   That makes the general form for the exponent a-n

To put all this together we get the formula for the nth derivative using the power rule as

n(x)=  a!/(a-n)!  x a-n



 I am not sure if this pattern was already discovered by someone else or not. I have done some research and have not been able to find anything about it online.

5 comments:

  1. This article is so great. Your points are well explained and you are using words in such a way that even non-native english speakers will understand. Thanks.

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    Replies
    1. Thank you for the comment, The fact that I am self taught in most of the math that I know contributes to my ability to explain things so most can understand. I am also bi-lingual so I understand how important it is to use words closely to how they are defined. I try to avoid "slang" as much as possible.

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  2. I admire the knowledge you show in this article. I just want to bring under the attention that there are many students that can not go to college to pursue a math degree because they they can't afford it. I stumbled upon a site that gives a list of scholarships and grants that are available for students to go back to school. Even if you are an adult with a job and want to complete your degree, going back to school grants are available to you to. I wish you success with a bright future!

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  3. Nice explanation, Higher order derivative is little difficult but interesting while solving them.It's a differential rule(power rule) in calculus.

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  4. People are always interested in general formulae for higher order derivatives. I would recommend you improve your knowledge of whats happening in the world of calculus :) . Most of all, keep it up!

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