I was recently explaining differentiation from first principles to a colleague and how differentiation can be used to obtain the tangent line to a curve at any point. While doing this, my colleague came back at me with an argument for which I had no satisfactory reply.
I was describing the tangent line to a curve at a specific point in the same way that I was taught at school - that it is a line that just touches the curve at that point and has gradient equal to the derivative of the curve at that point. My colleague then said that for a cubic curve, the line can touch the curve again at other points so I explained the concept again but restricted to a neighbourhood about the point in question.
He then came back with the argument of this definition when the "curve" in question is a straight line. He argued that in this case the definition of the tangent line as "just touching the curve at that point" is simply not true as it is coincident with the line itself and so touches at all points.
I had no comeback to this argument at all and had to concede that I should have just defined the tangent as the line passing through the point on the curve that has gradient equal to the derivative at that point.
Now this whole exchange left me feeling rather stupid as I hold a Phd in Maths myself and I could not adequately define a tangent without using the notion of differential calculus - and yet when I was taught calculus at school it was shown as a tool to calculate the gradient of a tangent line and so this becomes a circular argument.
I have given this serious thought and can find no argument to counter my colleagues observation of the inadequacy of the informal definition in the case when the curve in question is already a straight line.
Also, if I do this again in future with another colleague how can I avoid embarrassment again? At what point did I go wrong here with my explanations? Should I have avoided the geometric view completely and gone with rate of changes instead? I am not a teacher but have taught calculus from first principles to many people over the years and would be very interested in how it should be done properly.