A basic doubt on derivatives and tangents I want to prove that the derivative of a function at a point is actually the slope of the tangent of the curve at that point (and vice-versa). Now, what definition of tangent should I use ? A line which touches the curve at exactly one point is not correct I think.  
 A: The existence of a derivative at some point means that the function is "locally" almost linear at that point.  This is a very fundamental concept that tends to get buried in all the fuss of learning to compute derivatives.
So we could think of the "tangent" line as being the line the function would be if it really were linear in some small neighborhood of the point, instead of "almost linear".  Continuing to think geometrically, we are constructing that tangent line by using secant lines, which are clearly defined, through intervals that are smaller and smaller around the point.   
Much of the power of the derivative is, precisely, in this approximation to linearity.  Anything you might want to prove about functions is usually quite easy if the function is linear.  Then when it is not linear, the derivative (if it exists) allows us to say -- well, it is almost linear, at least in a small enough neighborhood of this point.  And a surprising number of proofs that work for linear functions can be extended to almost linear functions.
A: There are functions for which a tangent exists, but which are not differentiable.  Consider $f(x)=x^{1/3}$ at $x=0$.  So you the existence of a tangent line does not imply differentiability. 
