Let $\map f x = e^{\cos x}$. Then: {{begin-eqn}} {{eqn | l = \frac \d {\d x} \map f x       | r = -\sin x \, e^{\cos x}       | c = Chain Rule for Derivatives }} {{eqn | ll= \leadsto       | l = \frac {\d^2} {\d x^2} \map f x       | r = -\sin x \frac \d {\d x} e^{\cos x} + e^{\cos x} \, \map {\frac \d {\d x} } {-\sin x}       | c = Product Rule }} {{eqn | r = \sin^2 x \, e^{\cos x} - \cos x \, e^{\cos x}       | c =  }} {{eqn | r = \paren {\sin^2 x - \cos x} e^{\cos x}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d^3} {\d x^3} \map f x       | r = \paren {\sin^2 x - \cos x} \frac \d {\d x} e^{\cos x} + e^{\cos x} \map {\frac \d {\d x} } {\sin^2 x - \cos x}       | c = Product Rule }} {{eqn | r = \paren {\sin^2 x - \cos x} \paren {-\sin x} e^{\cos x} + e^{\cos x} \paren {2 \cos x \sin x + \sin x}       | c =  }} {{eqn | r = \paren {-\sin^3 x + 3 \cos x \sin x + \sin x} e^{\cos x}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d^4} {\d x^4} \map f x       | r = \paren {-\sin^3 x + 3 x \cos x \sin + \sin x} \frac \d {\d x} e^{\cos x} + e^{\cos x} \map {\frac \d {\d x} } {-\sin^3 x + 3 \cos x \sin x + \sin x}       | c = Product Rule }} {{eqn | r = \paren {-\sin^3 x + 3 \cos x \sin x + \sin x} \paren {-\sin x} \, e^{\cos x} + e^{\cos x} \paren {-3 \cos x \sin^2 x + 3 \paren {\cos^2 x - \sin^2 x} + \cos x}       | c =  }} {{eqn | r = \paren {\sin^4 x - 3 \cos x \sin^2 x - \sin^2 x - 3 \cos x \sin^2 x + 3 \paren {\cos^2 x - \sin^2 x} + \cos x} e^{\cos x}       | c =  }} {{eqn | r = \paren {\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x} e^{\cos x}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d^5} {\d x^5} \map f x       | r = \paren {\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x} \frac \d {\d x} e^{\cos x}       | c = Product Rule }} {{eqn | o =        | ro= +       | r = e^{\cos x} \, \map {\frac \d {\d x} } {\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x}       | c =  }} {{eqn | r = \paren {\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x} \paren {-\sin x} \, e^{\cos x}       | c =  }} {{eqn | o =        | ro= +       | r = e^{\cos x} \paren {4 \cos x \sin^3 x - 6 \paren {2 \cos^2 x \sin x - \sin^3 x} - 6 \cos x \sin x - 8 \cos x \sin x - \sin x}       | c =  }} {{eqn | r = \paren {-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x} e^{\cos x}       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d^6} {\d x^6} \map f x       | r = \paren {-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x} \frac \d {\d x} e^{\cos x}       | c = Product Rule }} {{eqn | o =        | ro= +       | r = e^{\cos x} \, \map {\frac \d {\d x} } {-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x}       | c =  }} {{eqn | r = \paren {-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x} \paren {-\sin x} \, e^{\cos x}       | c =  }} {{eqn | o =        | ro= +       | r = e^{\cos x} \paren {-5 \cos x \sin^4 x + 10 \paren {3 \cos^2 x \sin^2 x - \sin^4 x} - 15 \paren {\cos^3 x - 2 \cos x \sin^2 x} - 15 \paren {\cos^2 x - \sin^2 x} - \cos x}       | c =  }} {{eqn | r = \paren {\sin^6 x - 15 \cos x \sin^4 x + 45 \cos^2 x \sin^2 x - 16 \sin^4 x - 15 \cos^3 x + 45 \cos x \sin^2 x - 15 \cos^2 x + 16 \sin^2 x - \cos x} \, e^{\cos x}       | c =  }} {{end-eqn}} By definition of Taylor series: :$\map f x \sim \displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ and so expanding about $\xi = 0$: {{begin-eqn}} {{eqn | l = e^{\cos x}       | r = \frac {x^0} {0!} e^{\cos 0} + \frac {x^1} {1!} \paren {-\sin 0} \, e^{\cos 0} + \frac {x^2} {2!} \paren {\sin^2 0 - \cos 0} e^{\cos 0}       | c =  }} {{eqn | o =        | ro= +       | r = \frac {x^3} {3!} \paren {-\sin^3 0 + 3 \cos 0 \sin 0 + \sin 0} e^{\cos 0} + \frac {x^4} {4!} \paren {\sin^4 0 - 6 \cos 0 \sin^2 0 + 3 \cos^2 0 - 4 \sin^2 0 + \cos 0} e^{\cos 0}       | c =  }} {{eqn | o =        | ro= +       | r = \frac {x^5} {5!} \paren {-\sin^5 0 + 10 \cos 0 \sin^3 0 - 15 \cos^2 0 \sin 0 + 6 \sin^3 0 - 15 \cos 0 \sin 0 - \sin 0} e^{\cos 0} + \cdots       | c =  }} {{eqn | o =        | ro= +       | r = \frac {x^6} {6!} \paren {\sin^6 0 - 15 \cos 0 \sin^4 0 + 45 \cos^2 0 \sin^2 0 - 16 \sin^4 0 - 15 \cos^3 0 + 45 \cos 0 \sin^2 0 - 15 \cos^2 0 + 16 \sin^2 0 - \cos 0} e^{\cos 0} + \cdots       | c =  }} {{eqn | r = e \paren {1 + 0 \times x - \frac {x^2} 2 + \frac {x^3} 6 \paren {-0 + 0 + 0} + \frac {x^4} {24} \paren {0 - 0 + 3 - 0 + 1} + \frac {x^5} {120} \paren {-0 + 0 - 0 + 0 - 0 - 0} + \frac {x^6} {720} \paren {0 - 0 + 0 - 0 - 15 + 0 - 15 + 0 - 1} + \cdots}       | c = Sine of Zero is Zero, Exponential of Zero, Cosine of Zero is One }} {{eqn | r = e \paren {1 + \frac {x^2} 2 - \frac {x^4} 6 - \frac {31 x^6} {720} + \cdots}       | c =  }} {{end-eqn}} {{qed}}