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# 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
2 years, 5 months ago

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Do you know how to prove the original statement is true? Staff · 2 years, 5 months ago