▁Y ▁is ▁a ▁continuous ▁map . ▁ ▁High er ▁order ▁derivatives ▁are ▁defined ▁induct ively ▁via ▁ ▁A ▁function ▁is ▁said ▁to ▁be ▁C k ▁if ▁D k F ▁: ▁U ▁x ▁X ▁x ▁X x ▁... ▁x ▁X ▁→ ▁Y ▁is ▁continuous . ▁ ▁It ▁is ▁C ∞ , ▁or ▁smooth ▁if ▁it ▁is ▁C k ▁for ▁every ▁k . ▁ ▁Properties ▁▁ ▁Let ▁X , ▁Y , ▁and ▁Z ▁be ▁Fr éch et ▁spaces . ▁ ▁Suppose ▁that ▁U ▁is ▁an ▁open ▁subset ▁of ▁X , ▁V ▁is ▁an ▁open ▁subset ▁of ▁Y , ▁and ▁F ▁: ▁U ▁→ ▁V , ▁G ▁: ▁V ▁→ ▁Z ▁are ▁a ▁pair ▁of ▁C 1 ▁functions . ▁ ▁Then ▁the ▁following ▁properties ▁hold : ▁▁ ▁Fund amental ▁theorem ▁of ▁calculus . ▁If ▁the ▁line ▁segment ▁from ▁a ▁to ▁b ▁lies ▁entirely ▁within ▁U , ▁then ▁. ▁ ▁The ▁chain ▁rule . ▁D ( G ▁o ▁F )( u ) x ▁= ▁D G ( F ( u )) DF ( u ) x ▁ ▁for ▁all ▁u ▁ ε ▁U ▁and ▁x ▁ ε ▁X . ▁ ▁Linear ity . ▁D F ( u ) x ▁is ▁linear ▁in ▁x . ▁ ▁More ▁generally , ▁if ▁F ▁is ▁C k , ▁then ▁D F ( u ){ x 1 ,..., x k } ▁is ▁mult il ine ar ▁in ▁the ▁x ' s . ▁ ▁Taylor ' s ▁theorem ▁with ▁remainder . ▁Suppose ▁that ▁the ▁line ▁segment ▁between ▁u ▁ ε ▁U ▁and ▁u