What is the difference between the slope of a line tangent to a general curve and a line tangent to a circle?
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.