テイラー (3x-10)^{4/3}

解

テイラー

(3 x - 10)^{\frac{4}{3}}

- 10 (- 1)^{\frac{1}{3}} \cdot 1 0^{\frac{1}{3}} + 4 (- 1)^{\frac{1}{3}} \cdot 1 0^{\frac{1}{3}} x + \frac{2 ^{\frac{1}{3}}}{( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{2} + \frac{2 ^{\frac{1}{3}}}{15 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{3} + \frac{1}{60 \cdot 1 0 ^{\frac{2}{3}} ( - 1 ) ^{\frac{2}{3}}} x^{4} + \dots

解答ステップ

テイラーの公式を適用する

= - 10 (- 1)^{\frac{1}{3}} \cdot 1 0^{\frac{1}{3}} + \frac{\frac{d}{d x} ( ( 3 x - 10 ) ^{\frac{4}{3}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( ( 3 x - 10 ) ^{\frac{4}{3}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ( 3 x - 10 ) ^{\frac{4}{3}} ) ( 0 )}{3 !} x^{3} + \dots

導関数を評価する

= - 10 (- 1)^{\frac{1}{3}} \cdot 1 0^{\frac{1}{3}} + \frac{4 ( - 1 ) ^{\frac{1}{3}} \cdot 1 0 ^{\frac{1}{3}}}{1 !} x + \frac{\frac{2 \cdot 2 ^{\frac{1}{3}}}{( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}}{2 !} x^{2} + \frac{\frac{2 \cdot 2 ^{\frac{1}{3}}}{5 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}}{3 !} x^{3} + \frac{\frac{2 ^{\frac{1}{3}}}{5 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}}}{4 !} x^{4} + \dots

改良

= - 10 (- 1)^{\frac{1}{3}} \cdot 1 0^{\frac{1}{3}} + 4 (- 1)^{\frac{1}{3}} \cdot 1 0^{\frac{1}{3}} x + \frac{2 ^{\frac{1}{3}}}{( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{2} + \frac{2 ^{\frac{1}{3}}}{15 ( - 1 ) ^{\frac{2}{3}} \cdot 5 ^{\frac{2}{3}}} x^{3} + \frac{1}{60 \cdot 1 0 ^{\frac{2}{3}} ( - 1 ) ^{\frac{2}{3}}} x^{4} + \dots