Question

If 𝑓(𝑥 + 𝑦) = 𝑓(𝑥)𝑓(𝑦) for all 𝑥 , 𝑦 ∈ R and 𝑓(5)= 2, 𝑓′(0) = 3, then using the definition of derivatives, find 𝑓′(5)

Answer 1

6

Answer 2

The functional equation $f(x+y) = f(x)f(y)$ implies $f(0)=1$. Using the definition of the derivative, $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Substituting the functional equation, $f'(x) = \lim_{h \to 0} \frac{f(x)f(h) - f(x)}{h} = f(x) \lim_{h \to 0} \frac{f(h) - 1}{h}$. Since $f(0)=1$, the limit $\lim_{h \to 0} \frac{f(h) - 1}{h}$ is $f'(0)$. Thus, $f'(x) = f(x)f'(0)$. For $x=5$, $f'(5) = f(5)f'(0) = 2 \times 3 = 6$.