[tex]\bf x^2+4y^2=1\\\\
-----------------------------\\\\
\textit{using implicit differentiation}
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2x+4\cdot 2y\cfrac{dy}{dx}=0\implies x+4y\cfrac{dy}{dx}=0\implies \boxed{\cfrac{dy}{dx}=\cfrac{-x}{4y}}\\\\
-----------------------------\\\\
\cfrac{dy^2}{dx^2}=\cfrac{-1\cdot 4y-(-x)4\frac{dy}{dx}}{(4y)^2}\impliedby \textit{using the quotient rule}
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\cfrac{dy^2}{dx^2}=\cfrac{-4y+4x\frac{dy}{dx}}{4^2y^2}\implies \cfrac{dy^2}{dx^2}=\cfrac{-y+x\frac{dy}{dx}}{4y^2}\impliedby \textit{common factor}
[/tex]
[tex]\bf \cfrac{dy^2}{dx^2}=\cfrac{-y+x\cdot \boxed{\frac{-x}{4y}}}{4y^2}\implies \cfrac{dy^2}{dx^2}=\cfrac{-y-\frac{x^2}{4y}}{4y^2}
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\cfrac{dy^2}{dx^2}=\cfrac{\frac{-4y^2-x^2}{4y}}{4y^2}\implies \cfrac{dy^2}{dx^2}=\cfrac{-4y^2-x^2}{4y}\cdot \cfrac{1}{4y^2}
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\cfrac{dy^2}{dx^2}=\cfrac{-4y^2-x^2}{16y^3}\iff -\cfrac{1}{4y}-\cfrac{x^2}{16y^3}\impliedby
\begin{array}{llll}
\textit{distributing the}\\
\textit{denominator}
\end{array}[/tex]
