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Question

If xyyx=ab, find dydx.

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Solution

xyyx=ab
Putting u=xy and v=yx, we get
uv=ab
Differentiating with respect to x, we get
dudxdvdx=0(1)
Now, u=xy
Taking logarithm on both sides, we get
logu=ylogx
Differentiating with respect to x, we get
1ududx=y×1x+logx×dydx
dudx=u[yx+logx×dydx]
dudx=xy[yx+logx×dydx]

Also, v=yx
Taking logarithm on both sides, we get
logv=xlogy
Differentiating with respect to x, we get
1vdvdx=x×1ydydx+logy×1
dvdx=v[xydydx+logy]
dvdx=yx[xydydx+logy]

From (1), we get
xy[yx+logx×dydx]yx[xydydx+logy]=0
[xylogxxyx1]dydx=yxlogyy×xy1
dydx=yxlogyy×xy1xylogxxyx1

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