20040814, 16:09  #1 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Unusual differentiation
How would you differentiate
y = x^( x ) ^(x^)xright up to infinity ? In other words y =X to the power of x to the power of xup to infinity ? Mally 
20040814, 16:20  #2 
May 2004
New York City
4198_{10} Posts 
The function is infinite for x > 1,
is not well defined for x <= 0, and is constantly 1 for 0 < x <= 1. Hence the derivative is only defined for 0 < x < 1, and equals zero there !! 
20040814, 22:11  #3  
Mar 2004
499 Posts 
Quote:
I'm not much into complex analysis, so I don't know all the places that this function converges. However, here is a hint on taking the derivative: Let f[x] := x^f[x], where x^f[x] is defined as x^f[x] := exp[f[x]*Log[x]], i.e. the number e=2.71828... to the power of f[x]*Log[x], where Log[x] is the natural log. Take the derivative, and solve for df/dx This has a fairly nice answer, and is related to the Lambert Wfunction. (I really don't know anything about it, though ) 

20040815, 16:16  #4 
May 2004
New York City
2×2,099 Posts 
^^ Oops. That post was incorrect.
If x == sqrt(2) for example, then x^x^x^... == 2 Hence the range of convergence is wider than given there, and the function is not constant in that range. 
20040816, 03:08  #5 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
I do know how to differentiate f(x) = x^x. Maybe that can help:
f(x) = x^x = (e^ln x)^x = e^(x*ln x) f '(x) = x*ln x * e^(x*ln x) = x*ln x * x^x = (x^x) * x * ln x Last fiddled with by jinydu on 20040816 at 03:09 
20040816, 03:45  #6 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Sorry, I made a mistake with my last post. The key property is:
If y = e^(f[x]), then dy/dx = f '(x) * e^(f[x]) So here's the corrected derivation: f(x) = x^x = (e^ln x)^x = e^(x*ln x) Let g(x) = x*ln x g '(x) = (x * 1/x) + (ln x) = 1 + ln x f '(x) = (1 + ln x) * e^(x*ln x) = (1 + ln x) * x^x Now, I'll try to differentiate f(x) = x^(x^x): f(x) = x^(x^x) = (e^ln x)^(x^x) = e^(ln x * x^x) Let g(x) = x^x * ln x g '(x) = (x^x * 1/x) + (ln x) * (1 + ln x) * x^x g '(x) = ([x^x]/x) + ([x^x * ln x] * [1 + ln x]) f '(x) = ([x^x]/x) + ([x^x * ln x] * [1 + ln x]) * e^(ln x * x^x) f '(x) = ([x^x]/x) + ([x^x * ln x] * [1 + ln x]) * x^(x^x) Someone may want to check this. Obviously x^(x^(x^x)) is likely to be even more complicated. 
20040818, 15:49  #7  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Unusual differentiation
Quote:
The Solution: Let Y = x^x^x^X to infinity. Therefore y =x^y  0 <x<=1 Taking logs logy =ylogx Hence log y/y=logx Hence dx/dy = [ y*(1/y) logy*1]/ (y^2) Quotient rule y not=0 Therefore dy/dx= (y^2)/ 1logy Now y tends to 1 in this range as given above Therefore dy/dx = 1 as log1 =0 This gives the slope as 1 (tan inverse 1=45*) i.e. m=1 Hence this is the eqn. of a straight line bisecting the angle (90*) in 1st. quadrant therefore eqn of line is y=x [y=mx+c and c=0] I remain open for further discussion on this problem Mally 

20040819, 14:40  #8 
Dec 2003
Hopefully Near M48
2×3×293 Posts 
I assume you're saying that f' (x) = x. If that is the case, then f(x) = 0.5x^2 + c, where c is some constant.
I learned in my math class that integration is the reverse of differentiation. Also, when you integrate a function, you get a whole family of functions, which differ from each other only by a constant. Thus, if your reasoning is correct, x^x^x^x^x... = 0.5x^2 + c. That would be a surprising result. 
20040819, 16:40  #9 
Bronze Medalist
Jan 2004
Mumbai,India
4004_{8} Posts 
Unusual differentiation
Thank you for your query. You are making the wrong assumption so naturally you get a conflicting answer! PLease reread my post. It is better to use leibniz's notation. Its less confusing. The diff. coeff. is 1 and not x Thus dy/dx =1 integrating y=x+c. put c=0 as the eqn is derived to be y=x Therefore y=x Now if we integrate once again we get integral ydx=(x^2)/2+c and put c=0 therefore Integral ydx =(x^2)/2. The limits we set at 0 to 1 now draw the line y=x The integral denotes the area of a triangle formed by points (0,0) (1,0) and (1,1). The area of this triangle is 1/2(base * height) =(1*1)/2=1/2 Compare with the integral above viz. (x^2)/2. Here x =1, so area =1/2. Therefore both results from (Calculus and coord geom) tally Q.E.D :confused I am open for further questions Mally 
20040819, 18:47  #10 
May 2003
5F2_{16} Posts 
The function x^{x^{x^{x...}}} converges (for real numbers) on the interval [e^{e}, e^{1/e}]. You might want to google the term "infinite exponential."
mfgoode, you seem to have differentiated log(x) wrong. Best, ZetaFlux 
20040821, 16:11  #11 
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Unusual diferentiation
Thank you for your astute observation on my derivation. Yes I left an 'x' out of differentiation. Since x =1 it does not affect the final conclusion. I will be highly obliged if you could elucidate on the two exponential limilts you have mentioned. What would the curve eqn. be and how did you arrive at the limits? It will be very interesting and valuable to me All the best, Mally 
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