A differentiable function f has the property that f(x + y) = f(x) + f(y) 1 − f(x)f(y) holds for all x, y in the domain of f. It is known also that lim x→ 0 f(x) = 0, lim x →0 f(x) x = 5. (i) Use the definition of the derivative of f to determine f ′ (x) in terms of f(x).

f'(x) = 5[1 + f²(x)]

Step-by-step explanation:

By definition, f'(x) = lim h => 0 {[f(x + h) - f(x)]/h}

This was used, together with the given limits:

lim x => 0 f(x) = 0

lim x => 0 f(x)/x = 5,

to determine the derivative of the given function:

f(x + y) = [f(x) + f(y)]/[1 - f(x)f(y)].

The workings are shown in the attachments.