Attempting to. The "to" is part of an infinitive.
You'd be understood saying "attempting of driving faster", but it's not natural.
For an example of a case where "of driving" sounds natural, try these:
What caused the accident? It was just a case of driving too carelessly.
Stunt driving is a kind of driving performed in the movies, with tricks and often intentional accidents.
A couple more, with examples of "of" + gerund.
No one suspected me of driving so fast because of my quiet manner, but they were wrong!
You can "suspect someone of " + gerund.
She was convicted of driving under the influence of alcohol.
"Convicted of" + a gerund is very natural.
Prepositions can be funny as you move from one language to another. In some cases they translate very differently. Some languages, such as the Slavic and Germanic, often skip "of" in favor of declining a noun in the Genitive case.