设f (x)在(0,+∞)内有定义,f′(1)=2,又对于任意的x,y∈(0,+∞)恒有f(xy)=yf(x)+xf(y).求f(x).
问题描述：
设f (x)在(0,+∞)内有定义,f′(1)=2,又对于任意的x,y∈(0,+∞)恒有f(xy)=yf(x)+xf(y).求f(x).
答
令x=y = 1得f(1) = 0令 y = 1/x得 0 = f(x) / x + x f(1/x) 所以 f(1/x) = -f(x) / x^2对x求导得yf'(xy) = yf'(x) + f(y)令y = 1/x得f'(1)/x = f'(x)/x + f(1/x) = f'(x)/x - f(x) / x^2代入f'(1) = 2得f'(x) - f(x)...