How can I find the derivative of the equation of an ellipse centered at origin using implicit differentiation?

Question

I've altered the equation of an ellipse to make it easier to differentiate: (b^2)(x^2) + (a^2)(y^2) = (a^2)(b^2)

From here, I must use implicit differentiation so that I can find the slope of the tangent line on the ellipse. I need to see it step by step so that I can learn how to do this correctly. Thanks.

Answer 1

you have b²x² + a²y² = a²b², and we assume y is a function of x, so differentiating,
(d/dx)(b²x² + a²y²) = (d/dx)(a²b²)
(d/dx)(b²x²) + (d/dx)(a²y²) = 0
2b²x + 2a²y(d/dx)(y) = 0
2b²x + 2a²yy' = 0
2a²yy' = -2b²x
y' = -b²x / a²y