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Implicit function in advanced mathematics
arctany/x=㏑√(x? +y? )=? ln(x? +y? ) export x on both sides:

(y/x)'/[ 1+(y/x)? ]=? (x? +y? )'/(x? +y? )

[(y' x-y)/x? ]/[ 1+(y/x)? ]=? (2x+2y y')/(x? +y? )

Y is a function of x, and the derivative should use the derivative rule of compound function.

(y' x-y)/(x? +y? )=(x+y y')/(x? +y? )?

y' x-y=x+y y'→y'(x-y)=x+y

y'=(x+y)/(x-y)