Differentiate f and find its domain. Show step by step please!
f(x)=lnlnln(x)
Thanks!
no big deal
f = ln (ln ln x)
f' = 1/(ln ln x) * (1/x * 1/ln x)
= 1/(x(lnx)(ln lnx))
what is the domain?
I was trying to solve the same question and couldn't find the domain. if someone can help me with it, I will be thankful.
what is x is negative or zero Tim ?
To differentiate the function f(x) = lnlnln(x), we can use the chain rule of differentiation.
Step 1: Start by identifying the composite functions within the given function, f(x). In this case, we have three layers of natural logarithm functions:
h(x) = ln(x) (innermost function)
g(x) = ln(h(x)) (middle function)
f(x) = ln(g(x)) (outer function)
Step 2: Find the derivative of each layer of the composite function.
First, find the derivative of the innermost function h(x) = ln(x):
h'(x) = 1/x
Then, find the derivative of the middle function g(x) = ln(h(x)):
g'(x) = h'(x) * (1/h(x)) = (1/x) * (1/ln(x))
Finally, find the derivative of the outer function f(x) = ln(g(x)):
f'(x) = g'(x) * (1/g(x)) = ((1/x) * (1/ln(x))) * (1/ln(g(x)))
Step 3: Simplify the expression obtained in step 2.
Substitute g(x) = ln(h(x)) into the expression:
f'(x) = ((1/x) * (1/ln(x))) * (1/ln(ln(h(x))))
Step 4: Determine the domain of the derivative function.
Since the natural logarithm function is only defined for positive values, the domain of f(x) = lnlnln(x) is restricted to positive values of x.
Therefore, the domain of f'(x) is also restricted to positive values of x, where x > 1.
In summary, the derivative of f(x) = lnlnln(x) is f'(x) = ((1/x) * (1/ln(x))) * (1/ln(ln(h(x)))), and its domain is x > 1.