= Directional Derivative = A '''directional derivative''' is a generalization of [[Calculus/PartialDerivative|partial derivatives]]. <> ---- == Description == A [[Calculus/PartialDerivative|partial derivative]] describes the rate of change in one variable given that all others are held constant. Concretely, the partial derivative of ''f(x,y)'' with respect to ''x'' is the rate of change in ''x'' while holding ''y'' constant. A '''directional derivative''' expresses the rate of change in all variables in a given direction. The given direction can be expressed as angle ''θ'', but really represents a [[Calculus/UnitVector|unit vector]] like ''u⃗ = cosθi + sinθj''. The [[Calculus/VectorOperations#Dot_Product|dot product]] therefore projects the rates of change in the direction of this unit vector. In the above bivariate example, the directional derivative is given by: ''D,,u,, f(x,y) = f,,x,,cosθ + f,,y,,sinθ'' where ''f,,x,,'' is the partial derivative of ''f'' with respect to ''x'' and ''f,,y,,'' is with respect to ''y''. More generically, take the dot product of ''u⃗'' and the [[Calculus/Gradient|gradient]]. ---- CategoryRicottone