Let f x + y = f x ⋅ f y for all x , y ∈ R and f x = 1 + x ϕ x log ⁡ 3 . If lim x → 0 ⁡ ϕ x = 1 , then f ′ x is equal to

Step-by-step solution

f ′ x = lim h → 0 f x + h − f x h = lim h → 0 ⁡ f x ⋅ f h - f x h   ∵ f x + y = f x ⋅ f y = f x lim h → 0 ⁡ f h - 1 h = f x lim h → 0 ⁡ 1 + h ϕ h log ⁡ 3 - 1 h   ∵ f x = 1 + x ϕ x log ⁡ 3 = f x log ⁡ 3 lim h → 0 ⁡ ϕ h = f x × log ⁡ 3 × 1   ∵ lim h → 0 ⁡ ϕ h = 1 = log ⁡ 3 f x