y=x^x^x^x...
take ln of both sides
ln y = ln(x^x^x^x... )
ln y = x ln(x^x^x^x... )
lny = x ln(y)
1 = x
well, that got me nowhere.
are the x's staggered upwards or on the same level?
e.g
532 = 5^8 = 390625
but (5^3)^2 = 5^6 ≠ 5^8
if y=x^x^x^x...
dy/dx=?
plz show working thanks got no ideal at all
5 answers
second last line should have been
532 = 5^9 = 1953125
532 = 5^9 = 1953125
upward...
let's try this:
let y = x^x
lny = xlnx
y'/y = x(1/x) + lnx
dy/dx = y(1 + lnx)
= x^x + x^x(lnx)
let u = x^x^x , the x's are staggered
ln u = x ln(x^x)
u'/u = x(x^x + x^x(lnx))/x^x) + ln(x^x)
argghhhh!! I was looking for a pattern.
What am I not seeing ????
let y = x^x
lny = xlnx
y'/y = x(1/x) + lnx
dy/dx = y(1 + lnx)
= x^x + x^x(lnx)
let u = x^x^x , the x's are staggered
ln u = x ln(x^x)
u'/u = x(x^x + x^x(lnx))/x^x) + ln(x^x)
argghhhh!! I was looking for a pattern.
What am I not seeing ????
Let's pick up at
lny = x ln(y)
1/y y' = ln(y) + x/y y'
y' (1/y - x/y) = ln(y)
y' = y*ln(y)/(1-x)
I'd suggest not looking too hard for a pattern. Pop over to wolframalpha.com and try entering
x^x, then x^x^x, then x^x^x^x, and so on. The derivatives shown have an interesting pattern, but I'd have a hard time generalizing it to a general function of just x.
lny = x ln(y)
1/y y' = ln(y) + x/y y'
y' (1/y - x/y) = ln(y)
y' = y*ln(y)/(1-x)
I'd suggest not looking too hard for a pattern. Pop over to wolframalpha.com and try entering
x^x, then x^x^x, then x^x^x^x, and so on. The derivatives shown have an interesting pattern, but I'd have a hard time generalizing it to a general function of just x.
1 answer