方向微分
F ( S ) = F 1 ( F 2 ( S ) ) {\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))} なら、 ∂ F ∂ S : T = ∂ F 1 ∂ F 2 : ( ∂ F 2 ∂ S : T ) {\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)} 。
f ( S ) = f 1 ( F 2 ( S ) ) {\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}}))} なら、 ∂ f ∂ S : T = ∂ f 1 ∂ F 2 : ( ∂ F 2 ∂ S : T ) {\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)} 。
f ( v ) = f 1 ( v ) f 2 ( v ) {\displaystyle f({\boldsymbol {v}})=f_{1}({\boldsymbol {v}})~f_{2}({\boldsymbol {v}})} なら、 ∂ f ∂ v ⋅ u = ( ∂ f 1 ∂ v ⋅ u ) f 2 ( v ) + f 1 ( v ) ( ∂ f 2 ∂ v ⋅ u ) {\displaystyle {\frac {\partial f}{\partial {\boldsymbol {v}}}}\cdot {\boldsymbol {u}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {v}}}}\cdot {\boldsymbol {u}}\right)~f_{2}({\boldsymbol {v}})+f_{1}({\boldsymbol {v}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {v}}}}\cdot {\boldsymbol {u}}\right)} 。
f ( v ) = f 1 ( f 2 ( v ) ) {\displaystyle f({\boldsymbol {v}})=f_{1}(f_{2}({\boldsymbol {v}}))} なら、 ∂ f ∂ v ⋅ u = ∂ f 1 ∂ f 2 ∂ f 2 ∂ v ⋅ u {\displaystyle {\frac {\partial f}{\partial {\boldsymbol {v}}}}\cdot {\boldsymbol {u}}={\frac {\partial f_{1}}{\partial f_{2}}}~{\frac {\partial f_{2}}{\partial {\boldsymbol {v}}}}\cdot {\boldsymbol {u}}} 。