# How do you find \frac{d^2y}{dx^2} by implicit differentiation where x^2y+xy^2=3x

How do you find $\frac{{d}^{2}y}{{dx}^{2}}$ by implicit differentiation where ${x}^{2}y+x{y}^{2}=3x$
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Stella Calderon
Differentiating
${x}^{2}y+x{y}^{2}=3x$
with respect to x we get
$2xy+{x}^{2}{y}^{\prime }+{y}^{2}+2xy{y}^{\prime }=3$
So we get
${y}^{\prime }=\frac{3-{y}^{2}-2xy}{{x}^{2}+2xy}$
for the second derivative we obtain
$y{}^{″}=\frac{\left(-2y{y}^{\prime }-2y-2x{y}^{\prime }\right)\left(x+2xy\right)-\left(3-{y}^{2}-2xy\right)\left(2x+2y+2x{y}^{\prime }\right)}{{\left({x}^{2}+2xy\right)}^{2}}$
Now plug the result for y' in this equation!
William Appel