So, I'm supposed to solve for y''' of the function, $x^2 + y^2 = 9$.
I was able to solve for the second order derivative using implicit differentiation, resulting in:
$y^{''} = (\frac{-y^{2}-x^{2}}{y^{3}})$
Now, I'm a little confused, as I'm not sure if my answer for the third order is correct. To calculate for the third order implicit derivative, will I just use the quotient rule? Doing so, I got:
$y^{'''} = (\frac{{y^{4}{y^{'}}+2xy^{3}}-3x^{2}y^{2}y^{'}}{y^{6}})$
Is this correct? Or can This still be simplified?
Thank you in advance!
Best Answer
OK so you have
$y^{''}=-(x^{2}+y^{2})/y^{3}$.
I agree. But we know that $x^{2}+y^{2}=9$ and so $y^{''}=-9/y^{3}=-9y^{-3}$.
Now another round of implicit differentiation and substituting back in for $y^{'}=-x/y$ gets you there :)