The IRS doesn't decide what's deductible, Congress does. The fact that an expense is large is not one of the considerations in determining whether it's deductible.
I wouldn't bother with an office in your home. The IRS takes a hard look at business use of your home, so it's likely that it would be a red flag to some extent. To make your car trips deductible you would have to meet two sets of requirements - business use of your home and principal place of business. They overlap, but they are not the same. There are other requirements besides exclusive use. I just mentioned that because I thought it was likely that you didn't meet that requirement. You can read the requirements for principal place of business in IRS Publication 587, at the following link.
Principal Place of Business
Note that one of the considerations is "the relative importance of the activities performed at each place where you conduct business." Tutoring is obviously performed primarily in the clients' homes, not in your home. Also look at the examples of what are considered administrative or management activities. Preparing lessons does not fit comfortably into that category. Claiming that your home is your principal place of business for tutoring is a stretch. If I were you, I wouldn't push it.
