nth derivative of f(x)=ln(1-0,5x)

[thetimetrick]

Hi guys! So I have this question I'm stuck on:

Show that for the function f(x)=ln(1-x/2)
f^n(x)=- (n-1)! / (2-x)^n , ∀ n≥ 1

I hope someone can point me in the right direction, I'm pretty new to this subject so I would really appreciate any help

 

[tkhunny]

Can you produce the first 2 or 3 derivatives?

 

[thetimetrick]

Can you produce the first 2 or 3 derivatives?

Here's what I have:
f'(x)=-(1/(2-x))
f''(x)=-(1/(2-x)²)
f'''(x)=-(2/(2-x)³)

It goes on like that, and fits f^n(x)=- (n-1)! / (2-x)^n , is that enough proof?

 

[Dr.Peterson]

Do you know how to do a proof by induction?

Or, in your class is the word "show" used to mean something less formal than "prove"?

Either way, if you can demonstrate how one derivative is derived from the previous one, that will help.

 

[Steven G]

Here's what I have:
f'(x)=-(1/(2-x))
f''(x)=-(1/(2-x)²)
f'''(x)=-(2/(2-x)³)

It goes on like that, and fits f^n(x)=- (n-1)! / (2-x)^n , is that enough proof?

Of course that is not enough. Be honest with yourself, do you see the coefficient in front being a factorial? If you do then I apologize but if you don't then you need to compute a few more derivative to see for yourself that the factorial is there. Of course you can do this by mathematical induction but probably have not learned that yet. Take some more derivative and then you will see the factorials and then can *say* that the pattern has been confirmed.

 

[thetimetrick]

Do you know how to do a proof by induction?

Or, in your class is the word "show" used to mean something less formal than "prove"?

Either way, if you can demonstrate how one derivative is derived from the previous one, that will help.

Hi! I did not know what proof by induction was, but after seeing your message and googling it, I managed to apply it here, thank you so much!!

 

[thetimetrick]

Of course that is not enough. Be honest with yourself, do you see the coefficient in front being a factorial? If you do then I apologize but if you don't then you need to compute a few more derivative to see for yourself that the factorial is there. Of course you can do this by mathematical induction but probably have not learned that yet. Take some more derivative and then you will see the factorials and then can *say* that the pattern has been confirmed.

Thank you Yeah you're right, I didn't know what mathemarical induction was, but now that I found out it's a thing that exists I managed to apply it!