## 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 (

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.