Nth Derivative of Polynomial (Part 2)

Let n be the number of differentiation and b be the exponent of the given term of a polynomial

1. If $n>b$, then the nth derivative of a given polynomial is 0.

Ex. Find the 5th derivative of ${ 2x }^{ 4 }$?

Solution: Since the given n is 5 and b is 4, it adheres that $n>b$, so use the statement 2, thus:

$\frac { { d }^{ 5 } }{ { dx }^{ 5 } } { 2x }^{ 4 }=0$

To prove that this statement is correct, let's use the repeated differentiation method.

$y={ 2x }^{ 4 }$

$\frac { dy }{ dx } ={ 2(4)x }^{ 4-1 }={ 8x }^{ 3 }$

$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ 8(3)x }^{ 3-1 }={ 24x }^{ 2 }$

$\frac { { d }^{ 3 }y }{ { dx }^{ 3 } } ={ 24(2)x }^{ 2-1 }={ 48x }^{ 1 }=48x$

$\frac { { d }^{ 4 }y }{ { dx }^{ 4 } } ={ 48(1)x }^{ 1-1 }={ 48x }^{ 0 }=48$

$\frac { { d }^{ 5 }y }{ { dx }^{ 5 } } =0$

Note: If we find the 6th, 7th or either the 1000th derivative of ${ 2x }^{ 4 }$, it will give also a derivative of 0 since n is greater than b.

Note by Merzel Mark Guilaran
6 years, 7 months ago

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Do you know how to prove the original statement is true?

Staff - 6 years, 7 months ago

No. I lack of proofs of it but for sure the statement is true because there are many times that I try to use some polynomial expressions and find the most higher order derivative to find the derivative of the given polynomial and the result is correct and adheres the given statement. I will try my best to prove this statement true. thanks

- 6 years, 7 months ago