remember {} with key is same as derivedStateOf

remember {}

@Composable
fun ATest() {
    val frequentlyChangedState = remember { mutableStateOf(0) }
    LaunchedEffect(Unit) {
        launch {
            delay(5000) // delay so you can attach with Layout Inspector if you don't trust logs enough.
            while (frequentlyChangedState.value < 10) {
                delay(200)
                frequentlyChangedState.value++
            }
        }
    }

    val valueToDisplay = remember(frequentlyChangedState.value) {
        frequentlyChangedState.value / 2
    }
    Timber.i("test_ ATest outer") // 1 initial composition + 10 recompositions

    ATestInner(valueToDisplay)
}

@Composable
fun ATestInner(value: Int) {
    Timber.i("test_ ATestInner")  // 1 initial composition + 5 recompositions
    Text(text = "value: $value")
}

@bentrengrove Thank you for your response and sorry for late reaction.

However I meant that this is a bit confusing:
The difference between remember(key) and derivedStateOf is in the amount of recomposition. derivedStateOf {} is used when your state or key is changing more than you want to update your UI.
As you can see by the example above we can use remember(key) and dependent composable will be updated only if the resulting value changes. This is really impressive ability of composables. And the expression inside remember(key) is calculated only if key changed.

I'm not quite sure I understand the statement With derivedStateOf it wouldn't even have to evaluate whether to skip it or not. It would be great if you can explain where exactly evaluation will be skipped?

In the following example //2 invokes as much as frequentlyChangedState changes:

val valueToDisplay2 = remember {
    Timber.i("test_ before derivedStateOf") // 1
    derivedStateOf {
        Timber.i("test_ inside derivedStateOf") // 2
        frequentlyChangedState.value / 2
    }
}

Well, looks like crucial distinction is that remember {} is Composable whereas derivedStateOf {} not and therefore a new State stream can be created even outside Composable.