`"If "y^(x)=x^(y)," then find "(dy)/(dx).`

1.	`"If "y^(x)=x^(y)," then find "(dy)/(dx).`
Answer» Correct Answer - `(y(x log y-y))/(x(y log x -x))` `y^(x)=x^(y)` `"or "log y^(x)= log x^(y)` `"or "x log y = y log x` `"or "x(1)/(y)(dy)/(dx)+log y xx1 =(y)/(x)+log x (dy)/(dx)` `"or "((x)/(y)-log x)(dy)/(dx)=(y)/(x)-log y` `"or "(dy)/(dx)=(y/x-logy)/((x)/(y)-log x)=(y(y-x log y))/(x(x- y log x))=(y(x log y - y ))/(x(y log x -x))`

Discussion

`"If "y^(1//m)=(x+sqrt(1+x^(2)))," then "(1+x^(2))y_(2)+xy_(1)` is (where `y_(r)` represents the rth derivative of y w.r.t. x)A. `m^(2)y`B. `my^(2)`C. `m^(2)y^(2)`D. none of these
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The slope of the tangent to the curve `(y-x^5)^2=x(1+x^2)^2`at the point `(1,3)`is.
If for `x (0,1/4),`the derivative of `tan^(-1)((6xsqrt(x))/(1-9x^3))`is `sqrt(x)dotg(x),`then `g(x)`equals:`(3x)/(1-9x^3)`(2) `3/(1+9x^3)`(3) `9/(1+9x^3)`(4) `(3xsqrt(x))/(1-9x^3)`A. `(3)/(1+9x^(3))`B. `(9)/(1+9x^(3))`C. `(3xsqrt(x))/(1-9x^(3))`D. `(3x)/(1-9x^(3))`
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If `f(theta) = sin(tan^(-1)(sintheta/sqrt(cos2theta)))`, where `-pi/4 lt theta lt pi/4,` then the value of `d/(d(tantheta)) f(theta)` is