I take my handicap elderly mother to all her docters appointments and hospital visits. Can I deduct mileage expenses?

Yes but only if you are claiming your mother as a dependent (as a qualifying relative) or medical dependent.

Unfortunately, you will only get a tax benefit from this medical expenses (19 cents per mile driven for medical purposes) if the total of all deductible medical expense is over the 10% AGI medical limitation (7.5% if over age 65) and only if you itemize

TurboTax will guide you on entering all of your medical expenses so include everything you paid out-of-pocket including prescriptions, glasses, dentist bills, etc.

Here is a link for more information about claiming medical expenses on your tax return

An individual must meet all 4 of these requirements in order to be considered your Qualifying Relative:

1. Not a Qualifying Child: The individual cannot be your Qualifying Child and cannot be someone else's Qualifying Child. They are a Qualifying Child if they meet all the requirements, whether or not they are claimed as a dependent
2. Relationship: The person must either have lived with you for the entire year as a member of the household (a person who is not actually related to you may meet the requirements in this way), or be related to you in one of the following ways: your child, stepchild, grandchild or other descendant of one of your children (or stepchildren or foster children), son-in-law, daughter-in-law, brother, sister, half brother, half sister, stepbrother, stepsister, brother-in-law, sister-in-law, parent, stepfather, stepmother, father-in-law, mother-in-law, grandparent, and, if related by blood, aunt, uncle, niece, or nephew. Remember that a child whom you legally adopted is always considered to be your child. Also note that, for the purposes of this requirement, divorce or death does not change any relationship which was established by marriage (e.g. son-in-law, daughter-in-law, etc.)
3. Gross Income: The person must have made less than $4,000 in gross income during 2016.
4. Support: You must have provided more than half of the individual's total support during the year.

You can include medical expenses you paid for an individual that would have been your dependent except that:

1. He or she received gross income of $4,050 or more in 2016,
2. He or she filed a joint return for 2016, or
3. You, or your spouse if filing jointly, could be claimed as a dependent on someone else's 2016 return

Click this link for more additional information about claiming elderly parents as dependents