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[parent] higher order derivatives of sine and cosine (Derivation)

One may consider the sine and cosine either as real or complex functions. In both cases they are everywhere smooth, having the derivatives of all orders in every point. The formulae

$\displaystyle \frac{d^n}{dx^n}\sin{x} = \sin{(x+n\!\cdot\!\frac{\pi}{2})}$
and
$\displaystyle \frac{d^n}{dx^n}\cos{x} = \cos{(x+n\!\cdot\!\frac{\pi}{2})},$
where $ n = 0,\,1,\,2,\,\ldots$ (the derivative of the $ 0^\mathrm{th}$ order means the function itself), can be proven by induction on $ n$. Another possibility is to utilize Euler's formula, obtaining
$\displaystyle \frac{d^n}{dx^n}\cos{x}+i\frac{d^n}{dx^n}\sin{x} = \frac{d^n}{dx^... ...pi}{2}} = \cos{(x+n\!\cdot\!\frac{\pi}{2})}+i\sin{(x+n\!\cdot\!\frac{\pi}{2})};$
here one has to compare the real and imaginary parts - supposing that $ x$ is real.



"higher order derivatives of sine and cosine" is owned by pahio.
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See Also: fractional differentiation, higher order derivatives, example of Taylor polynomials for $\sin x$, cosine at multiples of straight angle


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Cross-references: real, imaginary parts, Euler's formula, induction, function, point, derivatives, smooth, complex functions, cosine, sine
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This is version 9 of higher order derivatives of sine and cosine, born on 2004-10-21, modified 2008-10-30.
Object id is 6395, canonical name is HigherOrderDerivativesOfSineAndCosine.
Accessed 3359 times total.

Classification:
AMS MSC: 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
 46G05 (Functional analysis :: Measures, integration, derivative, holomorphy :: Derivatives)
 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
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