if y=x^x^x^x...
dy/dx=?
plz show working thanks got no ideal at all
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
second last line should have been
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'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.
To find the derivative of y = x^x^x^x..., let's break it down step by step.
Step 1: Rewrite the expression
Let's define a new variable, say z, such that z = x^x^x^x...
Step 2: Take logarithm on both sides
Taking the logarithm will help simplify the expression. So, we can take the natural logarithm (ln) on both sides:
ln(z) = ln(x^x^x^x...)
Step 3: Apply logarithmic properties
Using the properties of logarithms, we can bring down the exponent to the front:
ln(z) = ln(x^(x^x^x...))
Step 4: Simplify the exponent
Since the exponent itself is a power tower, we can simplify it by substituting z back into the expression:
ln(z) = ln(z)
Step 5: Take the derivative implicitly
Now, differentiate both sides of the equation with respect to x, treating z as a function of x:
1/z * dz/dx = 1
Step 6: Solve for dz/dx
Rearrange the equation to solve for dz/dx:
dz/dx = z
Step 7: Substitute back for z
Replace z with x^x^x^x... to obtain the final derivative:
dz/dx = x^x^x^x...
Therefore, the derivative of y = x^x^x^x... with respect to x is dy/dx = x^x^x^x...
Note: This function represents an infinite power tower, and its derivative is nontrivial to compute analytically. However, the derivative expression itself provides the answer.