泰勒 ((e^t))/(1+t)

解答

泰勒

\frac{( e ^{t} )}{1 + t}

1 + \frac{1}{2} t^{2} - \frac{1}{3} t^{3} + \frac{3}{8} t^{4} - \frac{11}{30} t^{5} + \dots

求解步骤

应用泰勒公式

= 1 + \frac{\frac{d}{d t} ( \frac{e ^{t}}{1 + t} ) ( 0 )}{1 !} t + \frac{\frac{d ^{2}}{d t ^{2}} ( \frac{e ^{t}}{1 + t} ) ( 0 )}{2 !} t^{2} + \frac{\frac{d ^{3}}{d t ^{3}} ( \frac{e ^{t}}{1 + t} ) ( 0 )}{3 !} t^{3} + \dots

求导数

= 1 + \frac{0}{1 !} t + \frac{1}{2 !} t^{2} + \frac{- 2}{3 !} t^{3} + \frac{9}{4 !} t^{4} + \frac{- 44}{5 !} t^{5} + \dots

整理后得

= 1 + \frac{1}{2} t^{2} - \frac{1}{3} t^{3} + \frac{3}{8} t^{4} - \frac{11}{30} t^{5} + \dots

\frac{d}{d x} (\frac{24}{( x + 1 ) ^{5}})

d er i v a t i v e \frac{1}{x}

a re a y = 2 x, y = 9 x^{2}

\frac{d}{d x} (- 18 x - \frac{17}{x})

\int (- 8 y^{3}) d y