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let function f(x) be such that f(x+y) = f(x)f(y) for all x , y . Show that f ‘ (x) exists and f ‘ (x) = f(x)

let function f(x) be such that 
f(x+y) = f(x)f(y) for all x , y . 
Show that f ‘ (x) exists and f ‘ (x) = f(x)

Grade:12

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago

Hello student,
Please find answer to your question
f(x+y) = f(x)f(y)
f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
f'(x) = \lim_{h\rightarrow 0}\frac{f(x)f(h)-f(x)}{h}
f'(x) = \lim_{h\rightarrow 0}f(x).\frac{f(h)-1}{h}
f'(x) = f(x).f'(0)
\frac{df(x)}{dx} = f(x).f'(0)
\frac{df(x)}{f(x)} = f'(0).dx
\int \frac{df(x)}{f(x)} = \int f'(0).dx
ln(f(x)) = f'(0)x+c
f(x+y) = f(x)f(y)
x=0, y=0
f(0+0) = f(0)f(0)
f(0) = 1
ln(f(0)) = f'(0).0+c
c = 0
ln(f(x)) = f'(0)x
f(x) = e^{f'(0)x}
f'(x) = e^{f'(0)x}.f'(0)
If
f'(0) = 1
f'(x) = f(x)
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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