Differentiate the function.

This is a composed function and we'll differentiate from
the last function to the first.

We'll re-write the
function:

f(x) = (ln
x)^1/9

The last function is the power function and the
first function is the logarithmic function.

The function
f(x) is the result of composition between u(x) = u^1/9 and v(x) = ln
x

u(v(x)) = (lnx)^1/9

u'(x) =
(u^1/9)' = (1/9)*u^(1/9 - 1)

u'(x)
=  (1/9)*u^(1-9)/9

u'(x) =
1/9*u^8/9

v'(x) = (ln x)' =
1/x

[u(v(x))]' = 1/9x*(ln
x)^8/9

We'll use the power property of
logarithms:

a*ln b = ln
(b^a)

f'(x) = 1/(ln
x)^8*9x/9

We'll simplify and we'll
get:

f'(x) = 1/(ln
x)^8x/9