# Implicit Differentiation $y''$

I'm trying to find $y''$ by implicit differentiation of this problem: $4x^2 + y^2 = 3$

So far, I was able to get $y'$ which is $\frac{-4x}{y}$

How do I go about getting $y''$? I am kind of lost on that part.

You have $$y'=-\frac{4x}y\;.$$ Differentiate both sides with respect to $x$:

$$y''=-\frac{4y-4xy'}{y^2}=\frac{4xy'-4y}{y^2}\;.$$

Finally, substitute the known value of $y'$:

$$y''=\frac4{y^2}\left(x\left(-\frac{4x}y\right)-y\right)=-\frac4{y^2}\cdot\frac{4x^2+y^2}y=-\frac{4(4x^2+y^2)}{y^3}\;.$$

But from the original equation we know that $4x^2+y^2=3$, so in the end we have

$$y''=-\frac{12}{y^3}\;.$$

• Shouldnt the answer be $\frac{-12}{y^3}$ Commented Jun 29, 2012 at 20:49
• Since $4x^2+y^2$ since x and y must satisfy the original equation Commented Jun 29, 2012 at 20:56
• @soniccool: Yes: I inadvertently dropped a factor of $4$ in one term near the end. It’s fixed now. Commented Jun 29, 2012 at 21:00
• Yes i was right!! Im learning! Commented Jun 29, 2012 at 21:01