Find the expression for the slope on the lower half of the circle y^2 + x^2 = 25.
2. Attempt at a solution.
The text says you get 2x + 2y(dy/dx) = 0.
I got this and then solved for dy/dx to get dy/dx = -2y - 2x.
Then, I substituted for y the x value-expression for the lower region, y = - sqrt(25 - x^2)
and I got dy/dx = -2x - 2(sqrt(25 - x^2)).
Now the text gets the answer in another way:
2x + 2y(dy/dx) = 0;
then, 2x + 2(sqrt(25 - x^2))dy/dx = 0;
then, dy/dx = -2x/2(sqrt(25 - x^2)) = -x/sqrt(25 - x^2).
I see what they did. But what's wrong with the way I did it? Are the two answers equivalent in some way that I don't see, or how is mine wrong?