Final answer:
To differentiate the function, we use the quotient rule. The domain of f(x) is all x-values where the function is defined, which is (4, e + 4) ∪ (e + 4, +∞) in interval notation.
Step-by-step explanation:
To differentiate the function f(x) = x / (1 − ln(x − 4)), we need to apply the quotient rule since the function is a ratio of two differentiable functions.
The quotient rule states that if we have a function g(x) = u(x)/v(x), then g'(x) = (u'(x)v(x) − u(x)v'(x))/(v(x))^2, where u(x) is the numerator and v(x) is the denominator.
Here, u(x) = x and v(x) = 1 − ln(x − 4). So, u'(x) = 1 and v'(x) = −1/(x − 4). Thus, the derivative f'(x) is:
f'(x) = (1*(1 − ln(x − 4)) − x(-1/(x − 4)))/(1 − ln(x − 4))^2
This simplifies to: f'(x) = (1 − ln(x − 4) + x/(x − 4))/(1 − ln(x − 4))^2
The domain of f(x) consists of all values of x for which the denominator, 1 − ln(x − 4), does not equal 0 and x − 4 is positive (since ln(x) is undefined for non-positive numbers). Therefore, the function is undefined when x − 4 ≤ 0, which implies x ≤ 4, and when ln(x − 4) = 1, which is when x − 4 = e (e being the base of natural logarithm). Therefore, x cannot be 4 or e + 4. Taking these into account, the domain of f(x) is (4, e + 4) ∪ (e + 4, +∞), when written in interval notation.