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April 18th, 2018, 12:26 AM   #1
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Consider the non-constant differentiable function f of one variable which obeys the relation f(x)/f(y) = f(x -y) and d(f(0)) = p and
d((f5)) = q then find d(f(-5))

We have f(-5)) = f( 0 -5) = f(0)/f(5)

So ,

d(f(-5)) = (d(f(0)).f(5) - d(f(5)).f(0))/( f(5))^2
= (p.f(5) -q.f(0))/(q^2)

f(0) = f(5 -5) = f(5)/f(5) = 1

Only question how to find f(5) . Please help

Last edited by arybhatta01; April 18th, 2018 at 01:25 AM.
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April 18th, 2018, 01:38 PM   #2
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WHY do you want to know that? The point of this exercise is that you can answer this without knowing what f(5) is! And, in fact, while the information given is enough to determine what df(-5) (or df(5)) is, it is not sufficient to determine what f(5) itself is. (In general knowing the derivative of f is not sufficient to determine f- many different functions have the same derivative.)

For example, $\displaystyle f(x)= e^{ax}$ satisfies $\displaystyle \frac{f(x)}{f(y)}= \frac{e^{ax}}{e^{ay}}= e^{a(x- y)}= f(x- y)$, for any a.
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