Let n be the number of derivatives taken and b be the exponent of the given term of the given polynomial.

- If \(n=b\), then the nth derivative of the given polynomial is

\(\frac { { d }^{ n } }{ { dx }^{ n } } { ax }^{ b }=a(n!)\)

Example: Find the 4th derivative of \(3{ x }^{ 4 }\)?

Solution: Since n is equal to b, let's use the third statement.Thus,

\(\frac { { d }^{ 4 } }{ { dx }^{ 4 } } { 3x }^{ 4 }=3(4!)=72\)

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

\(y={ 3x }^{ 4 }\\ \\ \frac { dy }{ dx } =3(4){ x }^{ 4-1 }=12{ x }^{ 3 }\\ \\ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } =12(3){ x }^{ 3-1 }=36{ x }^{ 2 }\\ \\ \frac { { d }^{ 3 }y }{ { dx }^{ 3 } } =36(2){ x }^{ 2-1 }=72x\\ \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } =72(1){ x }^{ 1-1 }=72\)

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