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Is it possible to add a library created in android-x as module in my project which is not yet migrated to android-x?

I am trying to add a new module to my project to use as a library. But the issue is the new module developed using android x components. Is it possible to use the new module with my current project without migrating to android x?.

Technically not possible.
The simple reason is that Gradle cannot build an android app having both android support libraries and androidx libraries without any conversion, this would cause a merge conflict on classes.
What happens behind the scene during a build
Library A
implementation 'com.android.support:appcompat-v7:28.0.0'
Library B
implementation 'com.android.support:appcompat-v7:26.0.0'
Because the classes inside both packages are the similar: Gradle merges them to form one.
Now imagine the scenario
What happens behind the scene during a build
Library A
implementation 'com.android.support:appcompat-v7:28.0.0'
Library B
implementation 'androidx.appcompat:appcompat:1.0.0'
Now gradle becomes confuse because they're both appcompat libraries but their classes differs, hence they cannot be merged so it throws an error.
Nevertheless users whose project are on androidx can still use android support referenced libraries with the help of Jetpack Jetifier tool which would convert the support references into corresponding androidx libraries during build.
Unless you want to develop you own tool naming it UnJetifier then the option for building both android support referenced libraries and androidx libs in a support project would be possible. Best advice would be to migrate your app to androidx cause you are missing out on lots of features.

androidx.lifecycle.DefaultLifecycleObserver not found

I am having a problem where my build cannot find one of the androidx classes, specifically:
androidx.lifecycle.DefaultLifecycleObserver
Here are my gradle deps:
annotationProcessor "androidx.lifecycle:lifecycle-compiler:2.0.0"
implementation "androidx.lifecycle:lifecycle-runtime:2.0.0"
implementation "androidx.lifecycle:lifecycle-extensions:2.0.0"
I don't have any other problems getting androidx classes, just this one.
I have tried to invalidate caches in android studio and restart with no luck. Also a build clean as well as a gradle sync.
I verified here:
https://developer.android.com/jetpack/androidx/migrate
The mapping of that class from the support library to androidx:
android.arch.lifecycle.DefaultLifecycleObserver ->
androidx.lifecycle.DefaultLifecycleObserver
Any ideas on why this is happening?

I think that's because you're missing one dependency:
implementation "androidx.lifecycle:lifecycle-common-java8:$lifecycle_version"
Get the latest version of lifecycle-common-java8 here
See the documentation:
If you use Java 8 Language, then observe events with
DefaultLifecycleObserver. To include it you should add
"androidx.lifecycle:lifecycle-common-java8:" to your
build.gradle file.

How to get the latest Room version?

Would you know where to get the latest version for Room ?
On this Android Studio page, it is quoted that the latest Room version is 2.1.0-alpha3 but when I put this version in my build.gradle file, the project cannot compile and the following error is shown :
ERROR: Failed to resolve: android.arch.persistence.room:runtime:2.1.0-alpha3
Show in Project Structure dialog
Affected Modules: app

it is quoted that the latest Room version is 2.1.0-alpha3
It is.
when I put this version in my build.gradle file, the project cannot compile and the following error is shown
Your version is fine. Your artifact is the problem.
All new libraries are AndroidX. Your choices are:
Stick with whatever version you are using right now and keep your android.arch.persistence.room:runtime artifact and its classes, or
Migrate to AndroidX, in which case you would use androidx.room:room-runtime as the artifact, and need to change your code to reference the androidx classes for Room and everything else

The thing is, if you'r using AndroidX then the latest is 2.1.0-alpha3, or, if you are using Support libraries then it's 1.1.1 or whatever is latest and that's what mentioned in the documentation. As mentioned by CommonsWare, it's the artifact, and to find out the mappings, see this mapping documentation. And AndroidX is:
The AndroidX library contains the existing support library and also
includes the latest Jetpack components.
Example using support libraries:
dependencies {
// Other libraries...
implementation 'android.arch.persistence.room:runtime:1.1.1'
annotationProcessor 'android.arch.persistence.room:compiler:1.1.1'
}

The solution is to check AndroidX when starting a new project in Android Studio, then in the App Module Build.Gradle file the following works perfectly :
def room_version = "2.1.0-alpha03"
implementation "androidx.room:room-runtime:$room_version"
kapt "androidx.room:room-compiler:$room_version" // use kapt for Kotlin

Library not adding its dependencies when used

I made a library, but when I tried to use it with implementation 'com.example:mylibrary:1.3.0' in my app's build.gradle, I keep getting an error saying the ConstraintLayout dependency (which the library uses but not the app) is not found. However it was explicitly defined in the library's build.gradle with implementation.
When I ran gradlew app:dependencies on the terminal, it shows that the library has no dependencies, even though it actually has 2. This seems to be the source of the problem, Gradle can't detect the dependencies of the library.
I didn't run into this problem for a while but when I decided to remove the ConstraintLayout dependency from my app an error appears during build.

When you're using implementation for the dependencies in your library, the project which is dependent with it will not see the dependencies. You need to use api instead of implementation.

Failed resolution of: Lcom/google/android/gms/location/LocationRequest

I am using gradle dependency of
implementation 'com.google.android.gms:play-services-location:15.0.1'
at runtime I get below error
java.lang.NoClassDefFoundError: Failed resolution of: Lcom/google/android/gms/location/LocationRequest;
I am trying to fetch lat long using LocationRequest and when I am using this code in a standalone project it works. And when I am trying to build a library with same code I get above error.
I have checked, both standalone project and library project have same dependencies and versions.

This maybe because of 2 reasons
1st possible reason
When you used this library directly you used implementation method in build.gradle.
So you can use it directly in simple app module.
When you move it to your library and use that location library using same implementation option that location library can be only used by your library. and can't be used by app module in which you have used your library
Try by replacing that implementation by api for location library like this
api 'com.google.android.gms:play-services-location:15.0.1'
For more details refer this post for implementation vs api - Here
2nd possible reason
As you mentioned in comments, its a runtime error
As per my opinion it maybe because of obfuscation by ProGuard
add that class to keep ProGuard rule like this
-keep class com.google.android.gms.location.** { *; }
So this will stop obfuscation of that Location Request class

The code is working now with below changes (I am still looking for an explnation)
in the library gradle edited to below dependency
compileOnly "com.google.android.gms:play-services-location:15.0.1"
and in the app's gradle file added below dependency
runtimeOnly 'com.google.android.gms:play-services-location:11.6.0'
my guess is, since "compileOnly" takes care of adding the dependency in library and "runtimeOnly" takes care of using that dependency the code works and is able to find LocationRequest class.
P.S adityakamble49 's answer in the thread also helped. Please try that as well as it might work for your case.