Cho hàm số \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm % aabaGaamiEaaGaayjkaiaawMcaaiabg2da9iaadIhadaahaaWcbeqa % aiaaikdaaaGccqGHsislcaWG4baaaa!3E44! f\left( x \right) = {x^2} - x\), đạo hàm của hàm số ứng với số gia \(\Delta x\) của đối số x tại x0 là

\(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHuo % arcaWG5bGaeyypa0ZaaeWaaeaacaWG4bWaaSbaaSqaaiaaicdaaeqa % aOGaey4kaSIaeuiLdqKaamiEaaGaayjkaiaawMcaamaaCaaaleqaba % GaaGOmaaaakiabgkHiTmaabmaabaGaamiEamaaBaaaleaacaaIWaaa % beaakiabgUcaRiabfs5aejaadIhaaiaawIcacaGLPaaacqGHsislda % qadaqaaiaadIhadaqhaaWcbaGaaGimaaqaaiaaikdaaaGccqGHsisl % caWG4bWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzkaaaabaGaey % ypa0JaamiEamaaDaaaleaacaaIWaaabaGaaGOmaaaakiabgUcaRiaa % ikdacaWG4bWaaSbaaSqaaiaaicdaaeqaaOGaeuiLdqKaamiEaiabgU % caRmaabmaabaGaeuiLdqKaamiEaaGaayjkaiaawMcaamaaCaaaleqa % baGaaGOmaaaakiabgkHiTiaadIhadaWgaaWcbaGaaGimaaqabaGccq % GHsislcqqHuoarcaWG4bGaeyOeI0IaamiEamaaDaaaleaacaaIWaaa % baGaaGOmaaaakiabgUcaRiaadIhadaWgaaWcbaGaaGimaaqabaaake % aacqGH9aqpdaqadaqaaiabfs5aejaadIhaaiaawIcacaGLPaaadaah % aaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamiEamaaBaaaleaaca % aIWaaabeaakiabfs5aejaadIhacqGHsislcqqHuoarcaWG4baaaaa!7BC7! \begin{array}{l} \Delta y = {\left( {{x_0} + \Delta x} \right)^2} - \left( {{x_0} + \Delta x} \right) - \left( {x_0^2 - {x_0}} \right)\\ = x_0^2 + 2{x_0}\Delta x + {\left( {\Delta x} \right)^2} - {x_0} - \Delta x - x_0^2 + {x_0}\\ = {\left( {\Delta x} \right)^2} + 2{x_0}\Delta x - \Delta x \end{array}\)

Nên \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacE % cadaqadaqaaiaadIhadaWgaaWcbaGaaGimaaqabaaakiaawIcacaGL % PaaacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeuiLdq % KaamiEaiabgkziUkaaicdaaeqaaOWaaSaaaeaacqqHuoarcaWG5baa % baGaeuiLdqKaamiEaaaacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaai % yBaaWcbaGaeuiLdqKaamiEaiabgkziUkaaicdaaeqaaOWaaSaaaeaa % daqadaqaaiabfs5aejaadIhaaiaawIcacaGLPaaadaahaaWcbeqaai % aaikdaaaGccqGHRaWkcaaIYaGaamiEamaaBaaaleaacaaIWaaabeaa % kiabfs5aejaadIhacqGHsislcqqHuoarcaWG4baabaGaeuiLdqKaam % iEaaaacqGH9aqpdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeuiL % dqKaamiEaiabgkziUkaaicdaaeqaaOWaaeWaaeaacqqHuoarcaWG4b % Gaey4kaSIaaGOmaiaadIhadaWgaaWcbaGaaGimaaqabaGccqGHsisl % caaIXaaacaGLOaGaayzkaaaaaa!74EE! f'\left( {{x_0}} \right) = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{{\left( {\Delta x} \right)}^2} + 2{x_0}\Delta x - \Delta x}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \left( {\Delta x + 2{x_0} - 1} \right)\)

Vậy \(% MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacE % cadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWfqaqaaiGa % cYgacaGGPbGaaiyBaaWcbaGaeuiLdqKaamiEaiabgkziUkaaicdaae % qaaOWaaeWaaeaacqqHuoarcaWG4bGaey4kaSIaaGOmaiaadIhacqGH % sislcaaIXaaacaGLOaGaayzkaaaaaa!4B61! f'\left( x \right) = \mathop {\lim }\limits_{\Delta x \to 0} \left( {\Delta x + 2x - 1} \right)\)