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​Q50. Let f (xy) = xf (y) + y f (x) for all x, y R, and f (x) be differentiable in (0 , ) then determine f (x).
 

Dear student
Given f(xy)=xf(y)+yf(x)Replacing x by 1 and y by x then we get xf(1)=0f(1)=0,x0   (x,yR+)Now, f'(x)=limh0f(x+h)-f(x)h=limh0fx1+hx-f(x)h=limh0xf1+hx+1+hxf(x)-f(x)h=limh0f1+hxhx+limh0f(x)x=f'(1)+f(x)xSo, f'(x)=f'(1)+f(x)xf'(1)=f'(x)-f(x)xf'(1)x=xf'(x)-f(x)x2f'(1)x=ddxf(x)xOn integrating w.r.t.x., and taking limit 1 to x,we havef(x)x-f(1)1=f'(1)(logx-log1)f(x)x-0=f'(1)logx    [As f(1)=0]Hence f(x)=f'(1)(xlogx)
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