泰勒 1/(sqrt(1-x))

解答

泰勒

\frac{1}{1 - x}

1 + \frac{1}{2} x + \frac{3}{8} x^{2} + \frac{5}{16} x^{3} + \frac{35}{128} x^{4} + \dots

求解步骤

应用泰勒公式

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

求导数

= 1 + \frac{\frac{1}{2}}{1 !} x + \frac{\frac{3}{4}}{2 !} x^{2} + \frac{\frac{15}{8}}{3 !} x^{3} + \frac{\frac{105}{16}}{4 !} x^{4} + \dots

整理后得

= 1 + \frac{1}{2} x + \frac{3}{8} x^{2} + \frac{5}{16} x^{3} + \frac{35}{128} x^{4} + \dots

\frac{d}{d x} (\frac{8 x}{3 - tan ( x )})

d er i v a t i v e f (x) = 7 x + 2

\frac{d}{d x} (- x^{2})

\frac{d}{d x} (sec^{2} (x) tan (x))

\frac{\partial}{\partial x} (ln (x + 7 y + 2 z))