<s>[INST] Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$? here's my question
If $y^2=ax^2+bx+c$ then prove that  $y^3[d^2y/dx^2]$  is a constant .
I have solved this using the conventional method, taking square root, differentiating w.r.t to x  and using chain and quotient rule
But can't think of some alternative smaller and more efficient method ??
Can you ?
Also can someone suggest me any alternative to quotient rule
I need something more non conventional, time saving and vastly
Applicable method .
 [/INST] Using implicit derivation:
$$
(y^2)^\prime = 2yy^\prime = 2ax+b \Rightarrow y^\prime=\frac{2ax+b}{2y}
$$
$$
(y^2)^{\prime\prime} = 2(y^\prime)^2+2yy^{\prime\prime} = 2a \Rightarrow y^{\prime\prime}=\frac{a-(y^\prime)^2}{y}
$$
Thus
\begin{align}
y^3y^{\prime\prime}=y^2(a-(y^\prime)^2) &= y^2(a-\left(\frac{2ax+b}{2y}\right)^2) \\ 
&= ay^2-\frac{1}{4}(2ax+b)^2 \\
&= (a^2x^2+abx+ac)-\frac{1}{4}(4a^2x^2+4abx+b^2) \\
&=  (a^2x^2+abx+ac)-(a^2x^2+abx+\frac{b^2}{4}) \\
&= ac - \frac{b^2}{4} \ \ \ \text{(which is constant)}
\end{align}
</s>