Hello, this is my first time posting so apologies if the format lacks clarity. Going from the first derivative of y (line $3$) to the second derivative (line $4$), it appears (to me) that though they have differentiated the right hand side 'w.r.t $x$' but swapped the dt and $dx$, effectively differentiating w.r.t $t$ instead? My question is, am I understanding it correctly? If so, is this always okay? I thought you weren't supposed to treat the derivatives as fractions.
Thank you for taking the time to read my question.
This is the essence of the chain rule: differentiating with respect to $x$ will give the same result as differentiating with respect to $t$ and then multiplying by $\frac{dt}{dx}$.
If that doesn't sound like the chain rule you know, view $t$ as function of $x$. Then we want to differentiate $f(t(x))$ with respect to $x$. Applying the chain rule, we get $f'(t(x)) \cdot t'(x)$. The $f'(t(x))$ term is $\frac{df}{dt}$, and the $t'(x)$ term is $\frac{dt}{dx}$.