But it seems you just need to give some examples that involve differentiability? So, example functions, or example word problems. One example that's probably not in your textbook is ray tracing (which has real world applications). So, to find how a ray of light bounces off an object, you can use derivatives to get the tangent line, hence the normal, hence you can get the reflected ray. And you can have your object be non-differentiable in some parts. (In practice for ray tracing, we could be using more advanced techniques from vector calculus/linear algebra, but it's the same idea).
Another example is using Newton's method to find square roots ( is differentiable, and thus locally linear).