Total derivative

The derivative of a quantity with respect to an input when that quantity depends on the input both directly and through other variables that also change.

Chain rule: A differentiation rule that expresses a composite function’s derivative through the derivatives of its component functions. The total derivative is the chain rule applied to every path by which the input affects the quantity.
Independent and dependent variables: Variables distinguished by whether their values are chosen as inputs or determined by relationships to other variables. A total derivative tracks dependent variables as an independent input changes.
Sensitivity analysis: The study of how changes in model inputs affect its outputs. Total derivatives measure output sensitivity while accounting for induced changes in intermediate variables.
Partial derivative: A derivative with respect to one variable while other independent variables are held fixed. It excludes mediated changes that the total derivative includes.
Multivariable chain rule: A chain rule for functions whose intermediate variables depend on one or more inputs. It gives the total derivative when several changing variables mediate the dependence.
Composite function: A function formed by applying one function to the output of another. Indirect dependence in a total derivative arises when the quantity is a composite function.
Comparative statics: The analysis of how equilibrium outcomes change when an underlying parameter changes. Total derivatives trace equilibrium responses through variables determined by the model.
Functional derivative: A derivative that describes how a functional changes when its input function is varied. It concerns variation of an entire function, rather than ordinary dependence among finitely many variables.
Jacobian matrix: A matrix of first-order partial derivatives that describes a vector-valued function’s local linear change. Multiplying Jacobians composes the component changes used to calculate a total derivative.
Implicit function theorem: A theorem that gives conditions under which an equation defines some variables locally as functions of others. It lets implicit dependencies be differentiated to find their contribution to a total derivative.

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