What is taylor e^{-x},1 ?

The answer to taylor e^{-x},1 is 1/e-1/e (x-1)+1/(2e)(x-1)^2-1/(6e)(x-1)^3+1/(24e)(x-1)^4+\ldots

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taylor e^{-x},1

taylor

e^{- x}, 1

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

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

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

Refine

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

\frac{\partial}{\partial x} ((cos (x))^{2})

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

\int csc (4 x) d x

\frac{d}{d x} (arcsinh (x))

What is taylor e^{-x},1 ?
The answer to taylor e^{-x},1 is 1/e-1/e (x-1)+1/(2e)(x-1)^2-1/(6e)(x-1)^3+1/(24e)(x-1)^4+\ldots