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keonlo
Group Title
Is e^x the only function that has a same derivative of itself? if yes, how to proof it?
 2 years ago
 2 years ago
keonlo Group Title
Is e^x the only function that has a same derivative of itself? if yes, how to proof it?
 2 years ago
 2 years ago

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shandelman Group TitleBest ResponseYou've already chosen the best response.0
f(x) = 0 has a derivative of 0.
 2 years ago

JingleBells Group TitleBest ResponseYou've already chosen the best response.0
Yes, but I think 0 is just a constant not a function, no? I believe the value of e is unique just like pi. But correct me if I'm wrong!
 2 years ago

FoolAroundMath Group TitleBest ResponseYou've already chosen the best response.1
\[\frac{df}{dx} = f\]\[\Rightarrow \frac{df}{f}=dx\]\[\Rightarrow\int \frac{df}{f} = \int dx\]\[\Rightarrow ln(f) = x+c \]\[\Rightarrow f(x) = e^{x+c}\]\[\Rightarrow f(x) = k.e^{x}\] Here, \(k\) is any real number. So, any function of the form \(f(x) = k.e^{x} \) has the same derivative as itself. (k=0 implies that f(x) = 0 also is a solution)
 2 years ago

JingleBells Group TitleBest ResponseYou've already chosen the best response.0
In this case the constant k is just a translation, and the exponential still has to have e as it's base, right?
 2 years ago

FoolAroundMath Group TitleBest ResponseYou've already chosen the best response.1
Yes the exponential has to have \(e\) as its base, and k can be any real number..
 2 years ago

JingleBells Group TitleBest ResponseYou've already chosen the best response.0
Thanks!
 2 years ago

shandelman Group TitleBest ResponseYou've already chosen the best response.0
I would argue that f(x) = 0 is completely a function despite the fact that it is constant. It's graph passes the vertical line test, does it not? I will grant you it's not particularly the most interesting function but it is a function nonetheless.
 2 years ago

JingleBells Group TitleBest ResponseYou've already chosen the best response.0
Well by definition it maps all points on to 0. So it is a manytoone function. but I always wondered what x=0 on the xy coordinates would be?
 2 years ago

shandelman Group TitleBest ResponseYou've already chosen the best response.0
f(x) = 0 is a horizontal line overlapping the xaxis. x=0 is a vertical line overlapping the yaxis.
 2 years ago

JingleBells Group TitleBest ResponseYou've already chosen the best response.0
Does it qualify as a function was what I was wondering.
 2 years ago

FoolAroundMath Group TitleBest ResponseYou've already chosen the best response.1
a function is defined by \(f: D \rightarrow R \) D is the domain R is the range such that all elements in D have a map to R and for \(x_{1}\) in D, \(f(x_{1})\) can have only one value in R. i.e. it must be one>one or many>one. f(x) = 0 satisfies this definition.
 2 years ago
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