Formula for nth Derivative of f(x)= 1/(1-x)^2?

[christen1289]

Homework Statement

Find the formula for the nth derivative of the equation f(x)= 1/(1-x)^2

Homework Equations

The Attempt at a Solution

I have no idea how to attempt this problem. I've tried finding derivatives in order to find a pattern but I can't seem to come up with a pattern that would help me to find a formula

 

[dynamicsolo]

Homework Statement

Find the formula for the nth derivative of the equation f(x)= 1/(1-x)^2

The Attempt at a Solution

I have no idea how to attempt this problem. I've tried finding derivatives in order to find a pattern but I can't seem to come up with a pattern that would help me to find a formula

Let's start by listing the derivatives you've found. To make this easier to deal with, you could write the function as

f(x) = (1-x)^(-2) and use the Chain Rule. What is f'(x)?

 

[christen1289]

f'(x)=2(1-x)^-3

 

[genneth]

Can you write that in terms of the original f? Does that help when you apply the derivatives again?

 

[christen1289]

By finding up to the fourth derivative I came up with this formula:

nth deriv of f= (n+1)(n!)(1-x)^-(n+2)

 

[dynamicsolo]

By finding up to the fourth derivative I came up with this formula:

nth deriv of f= (n+1)(n!)(1-x)^-(n+2)

Yes! (I had to revise something I was going to say: the (-1) factor from the Chain Rule keeps canceling the minus sign from the exponent-factor, so this does stay positive.)

The one further simplification you can make is that (n+1) · (n!) = (n+1)!