Is there any way to prevent the hair on your face/chin from growing faster?

Yeah I would definitely recommend getting a new razor. Why haven't you gotten a new one?
If that still doesn't work you could always try those waxing strips. Those are good for small areas of skin (like hair above the lip). It'll hurt like hell though but I've heard they keep you hairless for at least a week or so.

Well, shaving actually makes it grow faster, but since you've already been shaving, there isn't much you can do to avoid that. One thing that can help some, is to thoroughly wash the area with cold water every morning and night, especially after the area has been exposed to heat. The cold will help shrink the pores, thus making the hair grow a little slower.

Buying a new razor, I would also recommend. Something with three to five blades would give you the closest shave, although they are a bit more expensive. But any new, fresh blade will give you the best results in a close shave that will keep it from growing back too soon.

In addition, when you shave, shave with water as warm as you can stand it to open the pores, and then wash it off with cold water to close them again.

Of course, results may vary depending on the person. This method has been pretty effective for myself, but I wish you luck and hope this helps.

A new razor would probably help.

However, the more often you shave, the quicker your facial hair grows back, that's why I tend to shave about once a week - even then, my facial hair tends to grow pretty fast.