1.

If `x^(y)+y^(x)=a^(b)` then find `(dy)/(dx)`.

Answer» Let `u=x^(y)andv=y^(x)`
`u=x^(y)`
`rArrlogu=logx^(y)=ylogx`
`rArr1/ucdot(du)/(dx)=y/x+logxcdot(dy)/(dx)`
`rArr(du/(dx)=u[y/x+logxcdot(dy)/(dx)]`
`=x^(y)[y/x+logxcdot(dy)/(dx)]`
`=ycdotx^(y-1)+x^(y)cdotlogxcdot(dy)/(dx)`
and `v=y^(x)`
`rArrlogv=logy^(x)=xlogy`
`rArr1/v(dv)/(dx)=x/ycdot(dy)/(dx)+logycdot1`
`rArr(dv)/(dx)=v(x/y(dy)/(dx)+logy)`
`=y^(x)(x/y(dy)/(dx)+logy)`
`=xcdoty^(x-1)cdot(dy)/(dx)+y^(x)logy`
Now `x^(y)+y^(x)=a^(b)`
`rArru+v=a^(b)rArr(du)/(dx)+(dv)/(dx)=0`
`rArrycdotx^(y-1)+x^(y)logxcdot(dy)/(dx)`
`+xcdoty^(x-1)(dy)/(dx)+y^(x)logy=0`
`rArr(dy)/(dx)(x^(y)logx+xcdoty^(x-1))=-(y^(x)logy+ycdotx^(y-1))`
`rArr(dy)/(dx)=(-(y^(x)logy+ycdotx^(y-1))/((x^(y)logx+xcdoty^(x-1))` Ans.


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