ðˈʌs wiː hæv ɐ fˈɔːɹmjʊlə fɔːɹ , wˈʌn ʌv wˈeɪz tə ɹˌɛpɹᵻzˈɛnt ðə dᵻɹˈɛkʃənəl dᵻɹˈɪvətˌɪv wˌɛɹ ɪz ˈɑːɹbɪtɹˌɛɹi; fɔːɹ sˈiː ɪts fˈʊl dˌɛfɪnˈɪʃən əbˈʌv, ɪts dᵻɹˈɛkʃənəl dᵻɹˈɪvətˌɪv wɪð ɹᵻspˈɛkt tʊ ˈɪswɛɹ ðə fˈɜːst tˈuː iːkwˈɔlᵻɾiz dʒˈʌst ʃˈoʊ dˈɪfɹənt ɹˌɛpɹᵻzˈɛntˈeɪʃənz ʌvðə dᵻɹˈɛkʃənəl dᵻɹˈɪvətˌɪv.
Thus we have a formula for , (one of ways to represent the directional derivative) where  is arbitrary; for  (see its full definition above), its directional derivative with respect to  iswhere the first two equalities just show different representations of the directional derivative.