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>bn>b, then the nth derivative of a given polynomial is 0.

Ex. Find the 5th derivative of 2x4{ 2x }^{ 4 }?

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

d5dx52x4=0\frac { { d }^{ 5 } }{ { dx }^{ 5 } } { 2x }^{ 4 }=0

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

y=2x4y={ 2x }^{ 4 }

dydx=2(4)x41=8x3\frac { dy }{ dx } ={ 2(4)x }^{ 4-1 }={ 8x }^{ 3 }

d2ydx2=8(3)x31=24x2\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ 8(3)x }^{ 3-1 }={ 24x }^{ 2 }

d3ydx3=24(2)x21=48x1=48x\frac { { d }^{ 3 }y }{ { dx }^{ 3 } } ={ 24(2)x }^{ 2-1 }={ 48x }^{ 1 }=48x

d4ydx4=48(1)x11=48x0=48\frac { { d }^{ 4 }y }{ { dx }^{ 4 } } ={ 48(1)x }^{ 1-1 }={ 48x }^{ 0 }=48

d5ydx5=0\frac { { d }^{ 5 }y }{ { dx }^{ 5 } } =0

Note: If we find the 6th, 7th or either the 1000th derivative of 2x4{ 2x }^{ 4 }, it will give also a derivative of 0 since n is greater than b.

Note by Merzel Mark Guilaran
4 years, 9 months ago

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1 vote

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

Calvin Lin Staff - 4 years, 9 months ago

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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

Merzel Mark Guilaran - 4 years, 9 months ago

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