First, when finding the magnitude of a vector, such as , don't square the unit vectors , , or , square only the components.
Regarding your actual question, you don't need to solve for . Like the position vector, the unit tangent vector varies from time to time, so naturally it is a function of time. Therefore, your answer of is correct (except for the "i" you placed after the in the denominator, which I removed).
Of course, your answer could be simplified. For example, the denominator is equal to . Assuming is positive, then .