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Differential Calculus> let function f(x) be such that f(x+y) = f...

question mark

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)

Drake , 10 Years ago

Grade 12

1 Answers

Jitender Singh

Last Activity: 10 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

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