The derivative of a smooth map is defined by composing derivations with pullback of functions. The formulas are more useful when each symbol has a job rather than merely decorating the theorem.
Notation
A smooth \(n\)-manifold is locally modeled on \(\mathbf R^n\) with smooth transition maps. A smooth map \(F:M\to N\) differentiates to linear maps between tangent spaces.
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
Stress the formula
Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.
Interpretation
What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.
Limit of the argument
Coordinates are computational tools, not intrinsic data. Tensorial formulas must transform correctly on chart overlaps.
A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.