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## Homework Statement

a) Find d

^{n}/dx

^{n}(x

^{n}) for n = 1,2,3,4,5. Give the general formula

b) Give the general formula for d

^{n+1}/dx

^{n+1}(x

^{n})

## Homework Equations

## The Attempt at a Solution

a) n = 1 : d/dx (x) = 1

n = 2 :d

^{2}/dx

^{2}(x

^{2}) = 2

n = 3 :d

^{3}/dx

^{3}(x

^{3}) = 6

n = 4 :d

^{4}/dx

^{4}(x

^{4}) = 24

n = 5 :d

^{5}/dx

^{5}(x

^{5}) = 120

Notice that there is a recurring pattern:

1,2,6,24,120→ multiply the derivative of the previous term by the existing term (2 times 1 = 2, 3 times 2 = 6, 4 times 6 = 24, 5 times 24 = 120)

This is where i'm a little confused, i know what the pattern is and it is a bit hard to explain, however i wrote this out as the general formula:

d

^{n}/dx

^{n}(x

^{n}) = n[d

^{n-1}/dx

^{n-1}(x

^{n-1})

Would this be considered right? Or is the general formula suppose to be an arithmetic sequence or a geometric sequence?

b) d

^{n+1}/dx

^{n+1}(x

^{n+1})

n = 1 :d

^{2}/dx

^{2}(x

^{1}) = 0

n = 2 :d

^{3}/dx

^{3}(x

^{2}) = 0

n = 3 :d

^{4}/dx

^{4}(x

^{3}) = 0

n = 4 :d

^{5}/dx

^{5}(x

^{4}) = 0

n = 5 :d

^{6}/dx

^{6}(x

^{5}) = 0

So then the general formula is:

d

^{n+1}/dx

^{n+1}(x

^{n+1}) = 0

Once again is this correct or would there be another sequence equation for this like the last one. I am fairly sure that there is not much of an issue of a sequence formula for part b because it will always result in zero but for part a im confused as to if my equation is correct and if it is not, what would the general formula be? Thank you.