Students learning calculus of several variables face a bewildering number of formulas if they try to learn the subject as a special formula for every case.  “The gradient is perpendicular to the tangent” is false for explicit surfaces, z=f[x,y], but true for implicit surfaces, F[x,y,z]=c.  It seems there are many equations for tangents.

However, if we are conscious of the explicit, implicit, parametric classification of equations and graphs, there is only one procedure for finding the equation of the tangent.  The unified procedure reduces the nonlinear problem to a linear one of the same type that locally approximates the original.  This is the analytical form of local linearity.

In the case of an explicit nonlinear equation and graph, the tangent equation procedure produces an explicit linear equation and graph.  The tangent procedure applied to an implicit nonlinear equation (or contour) and graph produces an implicit linear tangent equation and graph.  The case of a nonlinear parametric equation and graph produces a linear parametric tangent equation and graph.  This works in various dimensions.  

In every case the same 3 steps produce the equation of the tangent: (1) compute the symbolic total differential. (2) evaluate the partial derivatives at the point of tangency. (3) change local coordinates back to x-y-z or plot in local coordinates.   Various computations are as follows:

The tangent to an explicit surface: Procedure 3.3.1

The tangent to an implicit curve: Procedure 4.2.1

The tangent to an implicit surface: Procedure 4.5.1

The tangent to a parametric curve: Procedure 9.6.1

The tangent to a parametric surface: Procedure 15.3.1

The links above give the three step procedure and specific analytical examples of each case. The computations give the analytical result of "zooming in" as in the microscopic local linearity section.