The "notational second derivative" is true only when the function 𝑓(𝑥) is of a very specific form 
pic.twitter.com/ZFTgHFcqxp
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y' = y² is a very popular example of a simple ODE whose general formula doesn't cover all solutions
oh goodness.. now i'm woried that there's an infinite number of solutions i could have missed
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Nah, it is still a 2 dimensions space of solutions and you still get existence and uniqueness if you specify initial conditions for f and f'. So, if I'm not mistaken, you're only missing the solutions where you have an initial condition with f'=0, ie. the constant solutions :)
- End of conversation
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