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If f (x/y)= f(x)/f(y) ,AA y, f (y)!=0 an...

If `f (x/y)= f(x)/f(y)` ,`AA y, f (y)!=0` and `f' (1) = 2`, find f(x) .

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`"We have "f((x)/(y))=(f(x))/(f(y))`
Differentiating w.r.t. x, keeping y as constant we get`f'((x)/(y))(1)/(y)=(f'(x))/(f(y))`
Putting x=y, we get
`f'(1)(1)/(x)=(f'(x))/(f(x))`
`rArr" "(f'(x))/(f(x))=(2)/(x)`
Intergrating both sides, we get
`log_(e)f(x)=2log_(e)x+log c`
`rArr" "f(x)=cx^(2)`
In (1), putting x = y = 1, we get f(1) = 1
`therefore" "f(x)=x^(2)`
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