Differentiate One Infinite Product: Strategies & Formulas

[the dude man]

How do you differentiate one? Is it possible? Any formulas?

 

[AKG]

Apply d/dx to the expression.

 

[hypermorphism]

One differentiates functions, not numbers. Do you mean you want to know the derivative of the identity function? Over what domain? Or do you want to know the derivative of the function that always returns 1?

 

[the dude man]

y'(x) = y(x)*M

M=
----
\ f' / f
/
----
Does that work?

 

[AKG]

No, that doesn't work. Change one of the f's to a g, then it should work.

 

[the dude man]

I can't figure this out can someone refer me to proof cause i can't find one.

 

[the dude man]

The derivative of a function in the form of an infinite product.
Anything would be nice.

 

[the dude man]

Bump. .

 

[matt grime]

Please don't bump. You asked how to differentiate an infinite product, then posted an ascii art diagram that indicates an infinite sum, not product. Which is it? If you take time to learn how to post latex here it will help you.

 

[3.14159]

equation for derivative of n product

A similar question came to me while finding the derivative of a function equal to the product of three differentiable functions here is my generalization to a function equal to an infinite product:

Let $$g_{n} = \prod^n_{k=1}f_{k}$$ and assume that $$f'_n$$ exists. Then by the product rule of differentiation and some clever factoring
$$g'_n = (\prod^n_{k=1}f_{k})\sum^n_{j=1}\frac{f'_j}{f_j}$$
which is even more simply written as
$$g'_{n} = g_{n}\sum^n_{j=1}\frac{f'_j}{f_j}$$.

To see how I get to this generalization I will work out the derivative for
$$g_{3} = \prod^3_{k=1}f_{k}=f_1f_2f_3$$.
First,
$$g'_{3}= f_2f_3f'_1 + f_1(f_2f_3)'= f_2f_3f'_1 + f_1(f_2f'_3+f_3f'_2)= f_2f_3f'_1+f_1f_2f'_3+f_1f_3f'_2$$
Then noticing the relationship between this form and the original equation,
$$g'_3=\frac{g_3}{f_1}f'_1+\frac{g_3}{f_2}f'_2+\frac{g_3}{f_3}f'_3= g_3(\frac{f'_1}{f_1}+\frac{f'_2}{f_2}+\frac{f'_3}{f_3})$$
which is what the general function predicted,
$$g'_3=g_3\sum^3_{j=1}\frac{f'_j}{f_j}$$

I know that this statement is true for specific values of n, but I’m not sure what features this function has when g is an infinite product of differentiable functions. I hope that this helps with the question at hand.