$e^{\ln[x+w]} = xw$
I hope we agree left side is equal to right side.
Finding partial derivative $w$ given right side is simply $x$
But when I used online partial derivative calculator and I tried to find partial derivative of $w$ given left function, it would outcome $1$.
https://www.symbolab.com/solver/partial-derivative-calculator
My question is, how is this possible? Should left side produce same result? I double checked that left side is equal to right side.
No, I don't agree that the left side is equal to the right side, since$$e^{\log(x+w)}=x+w\ne xw.$$And, yes,$$\frac\partial{\partial w}e^{\log(x+w)}=\frac\partial{\partial w}(x+w)=1.$$