taylor 1/(2x),\at 2

Lösung

taylor

\frac{1}{2 x}, at 2

\frac{- \frac{1}{2 ( at 2 ) ^{2}}}{1 !} (x - at 2) + \frac{\frac{1}{( at 2 ) ^{3}}}{2 !} (x - at 2)^{2} + \frac{- \frac{3}{( at 2 ) ^{4}}}{3 !} (x - at 2)^{3} + \frac{\frac{12}{( at 2 ) ^{5}}}{4 !} (x - at 2)^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= \frac{1}{2 at 2} + \frac{- \frac{1}{2 ( at 2 ) ^{2}}}{1 !} (x - at 2) + \frac{\frac{1}{( at 2 ) ^{3}}}{2 !} (x - at 2)^{2} + \frac{- \frac{3}{( at 2 ) ^{4}}}{3 !} (x - at 2)^{3} + \frac{\frac{12}{( at 2 ) ^{5}}}{4 !} (x - at 2)^{4} + \dots

Fasse zusammen

= \frac{- \frac{1}{2 ( at 2 ) ^{2}}}{1 !} (x - at 2) + \frac{\frac{1}{( at 2 ) ^{3}}}{2 !} (x - at 2)^{2} + \frac{- \frac{3}{( at 2 ) ^{4}}}{3 !} (x - at 2)^{3} + \frac{\frac{12}{( at 2 ) ^{5}}}{4 !} (x - at 2)^{4} + \dots

\frac{d y}{d x} = - y x^{2}

\frac{d}{d x} (- (x^{2} + 3 x - 2)^{3})

p a r i t y y = x sin (x) + cot (x)

\frac{d}{d x} (\frac{( x + 1 ) ^{2}}{( x - 1 ) ^{2}})

d er i v a t i v e y = \frac{x + 2}{x - 2}