jˈuːzɪŋ ðɪ əbˌʌv pɹˈuːf, wiː nˈoʊ ɪz wˈɛldᵻfˈaɪnd ænd dˈɪfɚɹˌɛnʃɪəbəl, ænd fɹʌm ðɪs fˈɔːɹmjʊlə wiː kæn jˈuːz ðə ɡɹˈeɪdiənt θˈiəɹəm tʊ ˈiːzili dᵻɹˈaɪv ðə wˈɛlnˈoʊn fˈɔːɹmjʊlə fɔːɹ kˈælkjʊlˌeɪɾɪŋ wˈɜːk dˈʌn baɪ kənsˈɜːvətˌɪv fˈoːɹsᵻz:.
Using the above proof, we know  is well-defined and differentiable, and  (from this formula we can use the gradient theorem to easily derive the well-known formula for calculating work done by conservative forces: ).