泰勒 (x-5)^{7/3}

解答

泰勒

(x - 5)^{\frac{7}{3}}

25 (- 1)^{\frac{1}{3}} \cdot 5^{\frac{1}{3}} - \frac{35 ( - 1 ) ^{\frac{1}{3}} \cdot 5 ^{\frac{1}{3}}}{3} x + \frac{14 ( - 1 ) ^{\frac{1}{3}} \cdot 5 ^{\frac{1}{3}}}{9} x^{2} + \frac{14}{81 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{3} + \frac{7}{1215 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{4} + \dots

求解步骤

应用泰勒公式

= 25 (- 1)^{\frac{1}{3}} \cdot 5^{\frac{1}{3}} + \frac{\frac{d}{d x} ( ( x - 5 ) ^{\frac{7}{3}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ( x - 5 ) ^{\frac{7}{3}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( x - 5 ) ^{\frac{7}{3}} ) ( 0 )}{3 !} x^{3} + \dots

求导数

= 25 (- 1)^{\frac{1}{3}} \cdot 5^{\frac{1}{3}} + \frac{- \frac{35 ( - 1 ) ^{\frac{1}{3}} \cdot 5 ^{\frac{1}{3}}}{3}}{1 !} x + \frac{\frac{28 ( - 1 ) ^{\frac{1}{3}} \cdot 5 ^{\frac{1}{3}}}{9}}{2 !} x^{2} + \frac{\frac{28}{27 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}}{3 !} x^{3} + \frac{\frac{56}{405 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}}{4 !} x^{4} + \dots

整理后得

= 25 (- 1)^{\frac{1}{3}} \cdot 5^{\frac{1}{3}} - \frac{35 ( - 1 ) ^{\frac{1}{3}} \cdot 5 ^{\frac{1}{3}}}{3} x + \frac{14 ( - 1 ) ^{\frac{1}{3}} \cdot 5 ^{\frac{1}{3}}}{9} x^{2} + \frac{14}{81 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{3} + \frac{7}{1215 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{4} + \dots

x \to 3 lim (\frac{2}{( x - 3 )})

s l o p e x^{2} + 4 x - 1

\frac{\partial}{\partial x} (2 x y + 5 y^{2} - 3 x y^{3})

x \to 1 lim (e^{5 x^{4} - 5 x})

\frac{\partial}{\partial x} (x^{3} y + z^{2} x - z y^{2})