Differentiability: We use the problem of tangent planes to motivate the multivariable definition of differentiability and analyze some examples of functions that are not differentiable.
Rules of Differentiation: We prove that the rules of differentiation are basically all the same as in single variable calculus, and calculate some gradients using those rules.
Directional Differentiation: Some functions which are not differentiable at some point nevertheless still have sensible tangent lines at those points. We use this observation to motivate defining directional differentiation as a distinct yet closely related concept to differentiation.