is there any general way/rule of integrating f(x)/f’(x)?

[KYS]

Hi,

I was bored and wanted to try doing this just for fun. I tried letting g’(x) = f(x) / f’(x). And then from there I got 1 / g’(x) = f’(x) / f(x) and then I multiplied both sides by g’’(x) (we’re of course assuming f is infinitely differentiable otherwise this wouldn’t be possible). After that I integrated both sides (using integration by parts) and ended up with ln|g’(x)| = 1 - ln|f’(x)| + x + c. From the looks of it though all this gets me is that f(x) = e^(x+c) which makes sense but aren’t there other functions that could potentially work here like x^2? why am I just getting an exponential as the only solution? Am I missing something or violating some kind of rule here? And if I am, is there an actual way of doing this without doing what I did?

Thanks for the help in advance.

 

[Steven G]

I am a bit confused at what you are saying. f(x) can be ANY function. So f(x)/f'(x) is determined by f(x), which is any function.

Now you decided to define g'(x) = f(x)/f'(x). Yes 1/g'(x) = f'(x)/f(x) ....

How are you concluding what f(x) must be? It can be any function!

 

[JeffM]

In case you do not see what Jomo is saying, let's take your suggested function and restrict it to x > 0.

[MATH]f(x) = x^2 \implies f'(x) = 2x \implies \dfrac{f(x)}{f'(x)} = \dfrac{x^2}{2x} = 0.5x.[/MATH]
[MATH]g'(x) = \dfrac{f(x)}{f'(x)} \implies g'(x) = 0.5x \implies g(x) = 0.25x^2 + C.[/MATH]
Does this relate to your answer?

[MATH]ln\{g'(x)\} = ln \left ( \dfrac{f(x)}{f'(x)} \right ) = ln(x^2) - ln(2x) =\\ 2ln(x) - ln(x) - ln(2) = ln(x) - ln(2) = ln(0.5x) \ \checkmark.[/MATH]But [MATH]ln(0.5x) \ne 1 - ln(2x) + x + c.[/MATH]
So you did something wrong somewhere.

 

[KYS]

I am a bit confused at what you are saying. f(x) can be ANY function. So f(x)/f'(x) is determined by f(x), which is any function.

Now you decided to define g'(x) = f(x)/f'(x). Yes 1/g'(x) = f'(x)/f(x) ....

How are you concluding what f(x) must be? It can be any function!

Figured out my mistake, thanks. I canceled out one of the f(x)’s at the end of equation. So I actually should’ve ended up with 1 - ln|f’(x)| + ln|f(x)| which in turn takes me right back where I started. Haha.

But is there actually a general rule to integrating f(x)/f’(x)?

 

[Steven G]

Figured out my mistake, thanks. I canceled out one of the f(x)’s at the end of equation. So I actually should’ve ended up with 1 - ln|f’(x)| + ln|f(x)| which in turn takes me right back where I started. Haha.

But is there actually a general rule to integrating f(x)/f’(x)?

Try it figure it out for yourself. Try U-Substitution, try using parts, etc. Please post back.