No Need To Lose Brain Cells| Easiest Method | f(xy)=yf(x)+xf(y)

Hope you enjoy this video :)

I will make at least 5 videos about Functional Equation in this month. This is the 4th.

Playlist:

[angelmendez-rivera]

Very simply, notice that f(x•y) = f(x)•y + x•f(y) everywhere is equivalent to f(x•y)/(x•y) = f(x)/x + f(y)/y everywhere. Let g(x) = f(x)/x everywhere, hence g(x•y) = g(x) + g(y). Since g is continuous, we know that g(x) = A•ln(x) everywhere, for some A in R. Therefore, f(x) = x•g(x) = A•x•ln(x). Therefore, f(x•y) = A•x•y•(ln(x) + ln(y)) = A•x•y•ln(x) + A•x•y•ln(y) = f(x)•y + x•f(y), as expected. Q. E. D.

[CipriValdezate]

The solution that came to my mind was f(x) = derivative of x, because derivative of xy is x times derivative of y plus y times derivative of x: f'(xy) = yf'(x) + xf'(y).