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

f(x/y)=f(x)/f(y) and f'(1)=2024 , then

Let f :R to R be a continous and differentiable function such that f (x+y ) =f (x). F(y)AA x, y, f(x) ne 0 and f (0 )=1 and f '(0) =2. Let f (xy) =g (x). G (y) AA x, y and g'(1)=2.g (1) ne =0 The number of values of x, where f (x) g(x):

let f(x) be the polynomial function. It satisfies the equation 2 +f(x)* f(y) = f(x) + f(y) +f(xy) for all x and y. If f(2) =5 find f|f(2)| .

If f(x) is a function satisfying f(x)-f(y)=(y^(y))/(x^(x))f((x^(x))/(y^(y)))AA x,y in R^(*) and f'(1)=1, then find f(x)

If f(x+y) = f(x) + f(y) + |x|y+xy^(2),AA x, y in R and f'(0) = 0 , then

f((x)/(y))=f(x)-f(y)AA x,y in R^(+) and f(1)=1, then show that f(x)=In x

If f is a differentiable function satisfying 2f(x)=f(xy)+f((x)/(y)),AA x,y in R^(+), f(1)=0 and f'(1)=(1)/(ln6), then f(7776) =