April 18th, 2018, 12:26 AM  #1 
Member Joined: Jul 2017 From: KOLKATA Posts: 30 Thanks: 2  Derivative
Consider the nonconstant 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. 
April 18th, 2018, 01:38 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 
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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