How can I generalize the derivative of a reciprocal function?

[Aspiring]

Hi guys, I need help generalizing the derivative of the reciprocal of the function μ'(×).

What I would to find is a series representation whereby I don't have to find any derivatives of the function but merely replace powers and orders of derivatives.

Leibniz's series expression for the nth derivative of the two functions doesn't help nor the sum for the nth derivative of the composite of two functions( I forgot the name of the mathematician who derived it).

Any help would be appreciated.

 

[Stephen Tashi]

generalizing the derivative of the reciprocal of the function μ'(×).

Before we go to far. Let's make sure you mean "reciprocal" and not "inverse function".

What I would to find is a series representation whereby I don't have to find any derivatives of the function.

If the series doesn't contain any expressions involving the derivatives of "the function", what is it allowed to contain? Try to state your problem clearly. As it reads now we have:

$f(x) = \frac{1}{\mu'(x)}$

and you want some sort of series representation for $f'(x)$.

Are you talking about a power series?

 

[Aspiring]

Not Quite; just a function that when I enter, say n=1 , as its argument it results in it's first derivative and so on. I don't want to calculate the function at points or find a power/Taylor series representation.

I have tried to generalize but I can't find a pattern, using the quotient rule to differentiate that is.

Any Help? :)

 

[Stephen Tashi]

when I enter, say n=1 , as its argument it results in it's first derivative and so on

Which function is "it's"? $f(x)$?
Show an example of what you tried -because you aren't expressing your goal clearly.

 

[Aspiring]

Okay, let me clarify.

I wish to find a formula that will give me the nth derivative of the reciprocal of κ'[x].

Just as the the nth derivative of the function e^ax is given by a^n(e^ax); I'am looking for the equivalent with the above function. Whether the function for the nth derivative is as a sum or series of products, I'm happy.

I hope that helps.

 

[JJacquelin]

Okay, let me clarify.
I wish to find a formula that will give me the nth derivative of the reciprocal of κ'[x].

Hi !
This clarify nothing since nobody knows what K'(x) is.

 

[Klungo]

Are you looking for some the nth derivative of some function 1/k'(x)?
It wouldn't really differ from the formula of some function 1/f(x).

 

[Aspiring]

Yes exactly! just k'(x) is not specified!

Surely it must be possible to find an nth derivative formula using the quotient rule?

 

[Boorglar]

So basically, you want to find the general term of this pattern:

1/f(x) , -1/f(x)^2 * f'(x) , 2/f(x)^3 * f'(x)^2 - 1/f(x)^2 * f''(x) , ... ?

(where I start with 1/f, its derivative, its second derivative, ... up to the n-th derivative).


Honestly I don't know whether a simple formula exists, although I've heard of Fa di Bruno's more general formula for the n-th derivative of f(g(x)), which I absolutely don't understand.