differentiation of ellipse equation
In Morris Kline's 'Calculus', he puts the ellipse equation in this form, b^2X^2+a^2y^2=a^2b^2, and says this is the best way to differentiate it; i did it thinking implicit differentiation and the product rule, but I'd get four terms on one side and two terms on the other side. He doesn't show how he did the implicit differentiation, but his result is -b^2x/a^2y, which seems pretty hard to get to from the above first form of the elipse equation.
I managed to find an old calc book that arrives at the derivative as shown above, but by means of the usual form of the ellipse equation x^2/a^2 + y^2/b^2 = 1; i would think to differentiate from here, you'd use the quotient rule(or, maybe he moves the denominators up by means of negative exponents; i admit to not having tried that yet, but still . . .), but this other calc book just implicitly differentiates the numerators and leaves the denominators alone, and then rearanges to get the derivative indicated by Morris Kline in the first paragraph above.
So, I have two questions! One, how to get the derivative of the ellipse equation from the from Morris Kline first indicates above, and two, why this other calc book gets the correct answer by doing implicit differentiation on the numerators alone?