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:
- f 1(x)= a x a-1
- f 2(x)= a (a-1) x a-2
- f 3(x)= a (a-1)(a-2) x a-3
- f 4(x)= a (a-1)(a-2)(a-3) x a-4
- f 5(x)= a (a-1)(a-2)(a-3)(a-4) x a-5
- a (a-1)
- a (a-1)(a-2)
- a (a-1)(a-2)(a-3)
- a (a-1)(a-2)(a-3)(a-4)
Now with the a in the exponent xa we see that the a reduces by the the derivative we are on for example the second derivative , f2(x), the exponent is x a-2 That makes the general form for the exponent x a-n
To put all this together we get the formula for the nth derivative using the power rule as
f 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.