The chain rule application was incorrect. However, the line after it was fine. But then, I never saw any use of the Pythagorean theorem, which is necessary to relate x and y. There is one step in which you seem to have regarded dx/dt and dy/dt as like terms?!? I would start the problem as follows:
x^2+y^2=25, and hence
x(dx/dt)+y(dy/dt)=0.
Also, A=(xy)/2, and therefore
dA/dt=(x(dy/dt)+y(dx/dt))/2.
You know dx/dt, so by virtue of an equation above, you should be able to figure out dy/dt. Once you have x, dx/dt, y, and dy/dt, you just plug into the expression for the derivative of the area, and you're done. I think you had most of the basic ideas correct, just some mechanics of solving it were incorrect.
Incidentally, what happens as y>0?
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