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# higher order derivatives of sine and cosine

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

$\frac{d^{n}}{dx^{n}}\sin{x}\;=\;\sin{(x+n\!\cdot\!\frac{\pi}{2})}$ |

and

$\frac{d^{n}}{dx^{n}}\cos{x}\;=\;\cos{(x+n\!\cdot\!\frac{\pi}{2})},$ |

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

$\frac{d^{n}}{dx^{n}}\cos{x}+i\frac{d^{n}}{dx^{n}}\sin{x}\;=\;\frac{d^{n}}{dx^{% n}}e^{{ix}}\;=\;e^{{ix}}i^{n}\;=\;e^{{ix+in\frac{\pi}{2}}}\;=\;\cos{(x+n\!% \cdot\!\frac{\pi}{2})}+i\sin{(x+n\!\cdot\!\frac{\pi}{2})};$ |

here one has to compare the real and imaginary parts – supposing that $x$ is real.

Related:

FractionalDifferentiation, HigherOrderDerivatives, ExampleOfTaylorPolynomialsForSinX, CosineAtMultiplesOfStraightAngle

Type of Math Object:

Derivation

Major Section:

Reference

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## Mathematics Subject Classification

26B05*no label found*46G05

*no label found*26A24

*no label found*

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