Using Derivative to give a general formula

[tmlrlz]

Homework Statement

a) Find dⁿ/dxⁿ (xⁿ) for n = 1,2,3,4,5. Give the general formula

b) Give the general formula for dⁿ⁺¹/dxⁿ⁺¹ (xⁿ)

Homework Equations

The Attempt at a Solution

a) n = 1 : d/dx (x) = 1
n = 2 :d²/dx² (x²) = 2
n = 3 :d³/dx³ (x³) = 6
n = 4 :d⁴/dx⁴ (x⁴) = 24
n = 5 :d⁵/dx⁵ (x⁵) = 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:
dⁿ/dxⁿ (xⁿ) = 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) dⁿ⁺¹/dxⁿ⁺¹ (xⁿ⁺¹)
n = 1 :d²/dx² (x¹) = 0
n = 2 :d³/dx³ (x²) = 0
n = 3 :d⁴/dx⁴ (x³) = 0
n = 4 :d⁵/dx⁵ (x⁴) = 0
n = 5 :d⁶/dx⁶ (x⁵) = 0
So then the general formula is:
dⁿ⁺¹/dxⁿ⁺¹ (xⁿ⁺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 I am confused as to if my equation is correct and if it is not, what would the general formula be? Thank you.

The Attempt at a Solution

 

[Dick]

I think everything you've said is right. In terms of writing a general formula, you might want to check out the definition of the 'factorial' function written "n!".

 

[Mark44]

Homework Statement

a) Find dⁿ/dxⁿ (xⁿ) for n = 1,2,3,4,5. Give the general formula

b) Give the general formula for dⁿ⁺¹/dxⁿ⁺¹ (xⁿ)

Homework Equations

The Attempt at a Solution

a) n = 1 : d/dx (x) = 1
n = 2 :d²/dx² (x²) = 2
n = 3 :d³/dx³ (x³) = 6
n = 4 :d⁴/dx⁴ (x⁴) = 24
n = 5 :d⁵/dx⁵ (x⁵) = 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:
dⁿ/dxⁿ (xⁿ) = 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?

The above looks fine. Another way to write the result is to use the factorial.

d²/dx² (x²) = 2 = 2!
d³/dx³ (x³) = 6 = 3!
d⁴/dx⁴ (x⁴) = 24 = 4!
d⁵/dx⁵ (x⁵) = 120 = 5!

Your sequence 1, 2, 6, 24, 120, ... is neither an arithmetic sequence, in which each term is some constant plus the previous term, or a geometric sequence, in which each term is some constant times the previous term.

b) dⁿ⁺¹/dxⁿ⁺¹ (xⁿ⁺¹)
n = 1 :d²/dx² (x¹) = 0
n = 2 :d³/dx³ (x²) = 0
n = 3 :d⁴/dx⁴ (x³) = 0
n = 4 :d⁵/dx⁵ (x⁴) = 0
n = 5 :d⁶/dx⁶ (x⁵) = 0
So then the general formula is:
dⁿ⁺¹/dxⁿ⁺¹ (xⁿ⁺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 I am confused as to if my equation is correct and if it is not, what would the general formula be? Thank you.