Unsure About this Substitutional Method

$\displaystyle

x=t^2$

$\displaystyle \frac{dx}{dy}=\frac{dx}{dt}\cdot\frac{dt}{dy}$

$\displaystyle \frac{dx}{dy}=2t\cdot\frac{dt}{dy}$

$\displaystyle \frac{dy}{dx}=\frac{1}{2t}\cdot\frac{dy}{dt}$

NOTE I am solving a Second Order Differential Equation via substitution, thus I shall differentiate again - using the product rule - below I have differentiated the two multiples of $\displaystyle \frac{dy}{dx}$ separately and then added them together in the end to form $\displaystyle \frac{d^2y}{dx^2}$

$\displaystyle \frac{d}{dx}(\frac{1}{2t})=-\frac{1}{2t^2}\cdot\frac{dt}{dx}$

$\displaystyle \frac{d}{dx}\frac{dy}{dt}=\frac{d^2y}{dt^2}\cdot\f rac{dt}{dx}$

$\displaystyle \frac{d^2y}{dx^2}=\frac{1}{2t}\cdot\frac{d^2y}{dt^ 2}\cdot\frac{dt}{dx}-\frac{1}{2t^2}\cdot\frac{dt}{dx}\cdot\frac{dy}{dt}$

Am I correct?

If not please do explain where and how I am wrong.

If I am correct, please reassure me on why I have done this, I don't fully understand the whole concept.

Thanks in advance