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Question

If yx+xy+xx=ab, find dydx.

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Solution

Given that yx+xy+xx=abPutting u=yx, v=xy and w=xx , we get u+v+w=abdudx+dvdx+dwdx=0 ...iNow, u=yx
Taking log on both sides,
log u=x log y

1ududx=xddxlog y+log yddxx using product rule1ududx=x1ydydx+log y×1dudx=uxydydx+log ydudx=yxxydydx+log y ...iiAlso, v=xy
Taking log on both sides,
log v=y log x

1vdvdx=yddxlogx+logxdydx1vdvdx=y1x+logxdydxdvdx=vyx+logxdydxdvdx=xyyx+logxdydx ...iiiAgain, w=xx
Taking log on both sides,
log w=x log x

1wdwdx=xddxlog x+log xddxx1wdwdx=x1x+logx1dwdx=w1+log xdwdx=xx 1+log x ...ivFrom i,ii,iiiand iv, we have yxxydydx+log y+xyyx+log xdydx+xx1+log x=0x.yx-1+xy.log xdydx=-xx1+log x-y.xy-1-yxlog y dydx=-yx log y+y.xy-1+xx 1+log xx.yx-1+xy log x

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