<s>[INST] Implicit derivative - $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Let $y$ be a function of $x$ determined by the equation 
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
Find $\Large\frac{dy}{dx}$ and $\Large\frac{d^2y}{dx^2}$
I've obtained $\Large\frac{dy}{dx} = \frac{-xb^2}{ya^2}$ and $\Large\frac{d^2y}{dx^2}=\frac{-b^2a^2y^2 - x^2b^4}{a^4y^3}$, but the book gives $\Large\frac{d^2y}{dx^2} = \frac{-b^2}{a^4y^3}$ (the answer of the first one is correct)
I don't know what I am doing wrong:
$\Large\frac{d^2y}{dx^2} = \frac{-b^2(ya^2) +xb^2 a^2 \frac{dy}{dx}}{a^4y^2}$. Substituting $\Large\frac{dy}{dx}$ I take that answer
Thanks in advance!
 [/INST] Your answer is correct, just unsimplified.  Note:
$$
-b^2a^2y^2-x^2b^4=-b^2(a^2y^2+b^2x^2)=-b^2
$$
The second equality holds when you cross multiply the equation for the ellipse.
</s>