Explore the differential calculus for algebraic curves in this advanced mathematics lecture. Extend Lagrange's approach to study tangent lines, tangent conics, and higher-order approximations for general curves of the form p(x,y)=0. Examine the lemniscate of Bernoulli in detail, learning to classify curve points as elliptic, parabolic, or hyperbolic based on the nature of the quadratic approximation. Discover how this strategy generalizes to investigate algebraic surfaces in three-dimensional space, opening the door to a more comprehensive differential study of algebraic curves.