What is the difference between the slope of a line tangent to a general curve and a line tangent to a circle?
The idea is basically the same, in that the tangent skims off both the circle and the curve.
For the circle, the line going from the centre to the point of tangency
is perpendicular to the tangent.
Also the tangent to the circle touches it at a single point for all tangents.
For a general curve, there is a possibility that the tangent at a particular point on the curve
may actually touch the curve again at some other point,
for instance on a cubic, the tangent at a local minimum or local maximum
does cut the curve again,
but it touches the curve at a single point "in the vicinity" of the point of tangency.
For functions of a single variable, eg "x", we differentiate the function
to get an equation for the slope of the tangent.
Circle equations are functions of 2 variables x and y unless you use "parametric" equations,
so geometric techniques are normally used for tangents to circles,
whereas we use differentiation for functions.