It's not a crazy idea. You can think of it as calc I derivatives are a narrative that starts with limits, works through secant / tangent lines and the difference quotient to arrive at the definition of a derivative, then (typically) gives you a whole set of derivative calculation rules without trying to rigorously prove why e.g. the quotient rule works the way it does.

So here, it's like you are watching a sequel, and this professor knows that the rules for calculation will be most frequently used going forward, so they want to get to that point quickly.