If f(x) = 4ln(5 x+ 4ln(x)), find f'( x ).
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)
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...
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)
To find the derivative of f(x) = 4ln(5x + 4ln(x)), we will use the chain rule.
The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of f(g(x)) with respect to x is given by the product of the derivative of f with respect to g(x) and the derivative of g(x) with respect to x.
Let's go through the process step by step:
Step 1: Identify the outer function and the inner function.
In this case, the outer function is ln(x), and the inner function is 5x + 4ln(x).
Step 2: Differentiate the outer function.
The derivative of ln(x) with respect to x is 1/x.
Step 3: Differentiate the inner function.
The derivative of 5x with respect to x is 5.
The derivative of 4ln(x) with respect to x can be found using the chain rule. Let's call the inner function u = x. Then, the derivative of ln(u) with respect to u is 1/u. So, the derivative of 4ln(x) with respect to x is 1/x * 4.
Step 4: Apply the chain rule.
Now we can apply the chain rule by multiplying the derivative of the outer function with respect to the inner function by the derivative of the inner function with respect to x.
f'(x) = (1/x) * (5 + 4/x)
Simplifying further, we can combine the terms:
f'(x) = (5 + 4/x^2)
Therefore, the derivative of f(x) = 4ln(5x + 4ln(x)) is f'(x) = 5 + 4/x^2.