What is taylor e^xsin(x),0 ?

The answer to taylor e^xsin(x),0 is x+x^2+1/3 x^3-1/30 x^5-1/90 x^6+\ldots

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taylor e^xsin(x),0

taylor

e^{x} sin (x), 0

x + x^{2} + \frac{1}{3} x^{3} - \frac{1}{30} x^{5} - \frac{1}{90} x^{6} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 0 + \frac{1}{1 !} x + \frac{2}{2 !} x^{2} + \frac{2}{3 !} x^{3} + \frac{0}{4 !} x^{4} + \frac{- 4}{5 !} x^{5} + \frac{- 8}{6 !} x^{6} + \dots

Refine

= x + x^{2} + \frac{1}{3} x^{3} - \frac{1}{30} x^{5} - \frac{1}{90} x^{6} + \dots

\frac{d}{d x} (e^{e} + 2 ln (1 + x^{2}) - 3^{x})

\int \frac{3 x}{6 x ^{2} + 9} d x

f (x) = - \frac{2 x - 10}{x - 3}, at x = 2

y = x^{2}, x = y^{2}, y = 1

\frac{d}{d x} (\frac{x ^{3}}{x ^{3}})

What is taylor e^xsin(x),0 ?
The answer to taylor e^xsin(x),0 is x+x^2+1/3 x^3-1/30 x^5-1/90 x^6+\ldots