 # Find y" by implicit differentiation. x^2+4y^2=4 Antinazius 2021-11-17 Answered
Find y" by implicit differentiation.
${x}^{2}+4{y}^{2}=4$
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${x}^{2}+4{y}^{2}=4$
Differentiate both sides with respect to x
$\frac{d\left({x}^{2}+4{y}^{2}\right)}{dx}=\frac{d\left(4\right)}{dx}$
Remember that the derivative of a constant is 0
$\frac{d\left({x}^{2}\right)}{dx}+\frac{d\left(4{y}^{2}\right)}{dx}=0$
The Chain Rule for differentiation
$\frac{d\left[f\left[g\left(x\right)\right]\right]}{dx}=\frac{d\left[f\left[g\left(x\right)\right]\right]}{d\left[g\left(x\right)\right]}\cdot \frac{d\left[g\left(x\right)\right]}{fx}$
$2{x}^{2-1}+\frac{d\left(4{y}^{2}\right)}{dy}\cdot \frac{dy}{dx}=0$
$2x+4\cdot 2{y}^{2-1}\frac{dy}{dx}=0$
Subtract 2x from both sides
$8y\cdot \frac{dy}{dx}=-2x$
Divide both sides by 8y
$\frac{dy}{dx}=\frac{-2x}{8y}=\frac{-x}{4y}$
Differentiate both sides with respect to x
The Quotient Rule for differentiation
$\frac{d}{dx}\left[\frac{d\left(x\right)}{g\left(x\right)}\right]=\frac{{f}^{\prime }\left(x\right)\cdot g\left(x\right)-f\left(x\right)\cdot {g}^{\prime }\left(x\right)}{{\left[g\left(x\right)\right]}^{2}}$
$y{}^{″}=\frac{{\left(-x\right)}^{\prime }\cdot \left(4y\right)-\left(-x\right){\left(4y\right)}^{\prime }}{{\left(4y\right)}^{2}}$
$y{}^{″}=\frac{\left(-1\right)\cdot \left(4y\right)-\left(-x\right){\left(4y\right)}^{\prime }}{16{y}^{2}}$
$y{}^{″}=\frac{-4y+4x{y}^{\prime }}{16{y}^{2}}$
Substitute ${y}^{\prime }=\frac{-x}{}$