Nth Derivative of Polynomial (Part 3)

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.

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

#Calculus

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$