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Question

If log(x2+y2)(0.5)=tan1(xy) then show that dydx=(yx)(y+x)

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Solution

log(x2+y2)
Letx2+y2=t 2x+dx+2ydy=dt
2x+2ydydx=dtdx
ddxlogt(ddtlogt)(dtdx)=(1t.12t)(2x+2yy)=12t/2(x+yy)1=1x2y2(x+yy) ......(2)
ddxtan1(xy)
Let xy=tx=yt
or dx=dy+ydt
1dydx+ydtdxdtdx=1dydxtyddxtan1=ddt(tan1t)dtdx=11+t2(1yty)=11+t2(1yty)=11+x2y2{1y(xy)y}=y2y2+x2{yy(x)y2} ---(1)
Equality (1)&(2)
yyxx2+y2=x+yyx2+y2yx=yx+yy=y(x+y)y=yxx+y

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