Question

If f(x) = 4ln(5 x+ 4ln(x)), find f'( x ).

Answer 1

let f(x) = y , then

y = 4ln(5x + 4y)

dy/dx = 4/(5x + 4y) ( 5 + 4dy/dx)
(dy/dx)(5x) + (dy/dx)(4y) = 20 + 16(dy/dx)

(dy/dx)(5x + 4y - 16) = 20
dy/dx = 20/(5x + 4y - 16)

Answer 2

at the end when you get dy/dx = 20/(5x+4y -16) ... what do you replace y with? in the beginning you have y = 4ln(5x+4y) the answer is only in terms of x...

Answer 3

Jason, I actually misread your question as
f(x) = 4ln(5 x+ 4f(x)),
so please ignore the above reply totally.

Here is the real answer, much easier than my wrong interpretation.

f '(x) = [4/(5x + 4lnx)](5 + 4/x)