The solution is not that difficult. Consider the general quadratic parameteric equation, it is:
Putting the origin into this system you need to have c=0 and f=0, making things a bit easier. Now putting the other two points in this system gives you a system of four equations with four unknowns (a,b,d,e). Solving this system gives you the equation of the curve, I got:
Do this calculation yourself and make sure I did not make a mistake. Now the derivative can be found using the chain rule like:
Using now the equation of the curve and putting everything in there you have for the slope:
Putting the values of 0, 1/2 and 1 in there you have your solution. I added a small picture to make things more clear but make sure you do this calculation and come back if you have any questions.