taylor 3^x

Lösung

taylor

3^{x}

1 + ln (3) x + \frac{ln ^{2} ( 3 )}{2} x^{2} + \frac{ln ^{3} ( 3 )}{6} x^{3} + \frac{ln ^{4} ( 3 )}{24} x^{4} + \dots

Schritte zur Lösung

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{ln ( 3 )}{1 !} x + \frac{ln ^{2} ( 3 )}{2 !} x^{2} + \frac{ln ^{3} ( 3 )}{3 !} x^{3} + \frac{ln ^{4} ( 3 )}{4 !} x^{4} + \dots

Fasse zusammen

= 1 + ln (3) x + \frac{ln ^{2} ( 3 )}{2} x^{2} + \frac{ln ^{3} ( 3 )}{6} x^{3} + \frac{ln ^{4} ( 3 )}{24} x^{4} + \dots

x \to 1 lim (- 4 x + 6)

\frac{d y}{d x} = e^{2 x + 3 y}

\frac{d}{d x} (\frac{1}{6} 5 (6 - 5 x^{2})^{7})

\int \frac{x}{3 x - 1} d x

\int x e^{- \frac{1}{10} x^{2}} d x