calculus – Third-order autonomous ODE – if $y’ = p$, $y” = p’p$, what is $y”’ = $?

This question has been in my mind for quite some time.

Suppose we have an autonomous ODE which we can solve by substituting:

$$y’ = p(y)$$. From there on, we know that $$y” = p’_y cdot y’ $$

which is $$y” = p’p$$

But what would the third derivative be? I tried a calculation and I got that

$$y”’ = p”p^2 + p’p^2$$ but I’m not sure if I’m right or wrong!