Implication or equivalence in differentiation and integration

Question

Say we have $y'=2x+5$. Then if we differentiate $y'$ we have $y''=2$, and if we integrate it we have $y=x^2+5x+C$. Which of the following arrows correctly connects these two equations?

$$y'=2x+5\Longleftrightarrow y''=2$$ or $$y'=2x+5\Longrightarrow y''=2$$

And $$y'=2x+5\Longleftrightarrow y=x^2+5x+C$$ or $$y'=2x+5\Longrightarrow y=x^2+5x+C$$

My guess is that for differenting $y'$ we should have the implication arrow, and for integrating we probably should have the equivalence arrow.

Answer 1

My guess is that for differenting $y′$ we should have the implication arrow, and for integrating we probably should have the equivalence arrow.

Why are you guessing?   Do you not have some reasoning, or at least intuition, for why this might be so?

$\implies$ indicates "is sufficient (but maybe not necessary) for".

•   That is: $A\implies B$ guarantees that $B$ will be true if $A$ is, but does not prohibit $B$ from being true when $A$ is not.   $A$ is enough to derive $B$, but may not be needed (having something else might do instead).

$\iff$ indicates "is sufficent and necessary for".

• So if you can assert that $A$ is necessary for $B$, as well as sufficient, use $A\iff B$.   If you can only be certain that $A$ is sufficient for $B$ use $A\implies B$.

Such as: $y''=2$ is derived from $y'= 2x+3$ but may also be derived from $y'=2x+A$ for any constant $A$ at all.   So having $y'=2x+3$ is sufficient but not necessary to derive $y''=2$.   Therefore $y'=2x+3\implies y''=2$ .