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if u=f(x/y,y/z,z/x),prove that xdu/dx+ydu/dy+zdu/dz

if u=f(x/y,y/z,z/x),prove that xdu/dx+ydu/dy+zdu/dz

Grade:12th pass

1 Answers

Sapna
15 Points
9 months ago
u=f(r,s,t) and r=x/y, s=y/x, t=z/x
du/dx= du/dr*dr/dx+du/ds*ds/dy+du/dt*dt/dx
du/dx=f'(u)(1/y)+0+f'(u)(-z/x²)
xdu/dx=f'(u)(x/y)+f'(u)(-z/x)
Similarly
ydu/dy=f'(u)(-x/y)+f'(u)(y/z)  and
xdu/dz=f'(u)(-y/z)+f'(u)(z/x)
Now
xdu/dx+ydu/dy+xdu/dz
Putting the value
=0 answer

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