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.

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

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

There are no comments in this discussion.