Let f (x), X belong to R, X is not equal to 0, for any non-zero real number x, y, satisfy f (XY) = f (x) + F (y), find f (1), f (- 1)

Let f (x), X belong to R, X is not equal to 0, for any non-zero real number x, y, satisfy f (XY) = f (x) + F (y), find f (1), f (- 1)

x=1 y=1
f(xy)=f(x)+f(y) f(1)=f(1)+f(1)
f(1)=0
x=-1 y=-1
f(1)=f(-1)+f(-1)=0
f(-1)=0

For all real numbers x, y function f (x) satisfies the condition f (XY) = f (x) f (y), tangent f (0) is not equal to zero, find f (x)

Let x = y = 0, then f (0) = f (0) squared
Because f (0) is not equal to 0, f (0) = 1
Let y = 0, then f (x) = 1

Piecewise function, f (x) = SiNx ^ 2 / 2x, X is not equal to zero, f (x) = 0, x = 0, find f (0) prime

F'(0)
=lim_ {x->0}(F(x)-F(0))/x
=lim_ {x->0}(sinx^2/2x-0)/x
=lim_ {x->0} sinx^2/2x^2
=(1/2)lim_ {x->0} sinx^2/x^2
=(1/2)lim_ {x->0} sinx/x
=1/2