kth Derivative of x^n

Date: 03/04/2003 at 05:27:14
From: Chris
Subject: What is a quick way to find the nth derivative of a function?

Example: the 7th derivative of x^8 ...

Date: 03/04/2003 at 11:55:22
From: Doctor Ian
Subject: Re: What is a quick way to find the nth derivative of a 
function?

Hi Chris,

In general, there's no shortcut.  You just have to take all the
derivatives.  

However, in some special cases, you can see right away (or from the
first few derivatives) what's going to happen.  For example, if

  f(x) = e^x

then 

  f'(x) = f''(x) = f'''(x) = ... = e^x

Or if 

  f(x) = sin(x)

then

  f'(x) = cos(x)
  
  f''(x) = -sin(x)

  f'''(x) = -cos(x)

  f''''(x) = sin(x)

so the (n+4)th derivative will be the same as the nth derivative. 

Let's look at your example:

  f(x) = x^8
  
  f'(x) = 8x^7

  f''(x) = (8)(7)x^6
   .
   .

In general, the kth derivative of x^n will be 

   [n!/(n-k)!] x^(n-k)

So the 7th derivative of x^8 will be 

   [8!/(8-7)!] x^(8-7) = [8!] x

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/