I believe that the answer to #2 is incorrect and in fact, the correct answer is not present.  Potential energy is the negative integral of the INTERNAL work over distance.  This is equivalent to the positive integral of the EXTERNAL work over the same distance.  For instance, as I apply a force upwards on an object to lift it up, the potential energy increases because force and displacement are in the same direction.  But, my applied force is an external force.  Gravity is the internal force and operates in the opposite direction.  A similar analysis may be done with springs.  Answers A, C and D are wrong because it transposes integral and derivative.

On another note, anytime you say derivative or integral, you should specify over what variable you are integrating or taking a derivative of.  In this case, position or displacement.  Most situations, we think of are with respect to time.

Michael, Thank you for pointing that out, and I changed it from negative to positive as it is an external force.
We haven't mentioned which variable - only because the variable might be x or y or r or any other value, and it seemed more straightforward to trust the person solving the problem to know which variable to use.

John

And, this actually belongs in the Energy unit!

Sorry.  I thought I was in the energy unit.  Not sure how I ended up here.  And why you understood me :D

Not a problem - I've been using NJCTL resources for almost 15 years - I hope I can recognize where these problems are!

John