The way I see it, you should have the machinery in your head, so that you can derive things if pressed, and you should have the ability to work largely in your head and actually ponder things, but you shouldn't necessarily memorize formulas.

For example, I haven't even memorized the formula for the quadratic equation, I just complete the square in my head every time.

I think it's important to end up knowing them and to end up being able to calculate with paper and pencil to some degree and because calculus is very basic it's something one should be able to do in that way.

You should be able to imagine a right angle triangle sitting on the hypotenuse of another right angled triangle and figure out the x and y coordinates of the corner and how to get the sum of angles formulas from that, or to in general imagine things from calculus, like how the product rule can move what what is differentiated from one function to another etcetera.