Đạo hàm hàm số đặc biệt |
Đạo hàm hàm số
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![{\displaystyle \Gamma (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec077ba0bdbf87c0d66173bc4d98598fe582ac37) |
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![{\displaystyle \Gamma (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec077ba0bdbf87c0d66173bc4d98598fe582ac37) |
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![{\displaystyle \zeta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67cfc06757b7c817532b8d83f29d9e5378cbcd55) |
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![{\displaystyle \zeta (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67cfc06757b7c817532b8d83f29d9e5378cbcd55) |
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Hàm số |
Đạo hàm bậc N
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![{\displaystyle y=F(G(x))\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bef313090aa35f6cb8152dd69c6e6a161ee898e) |
where
and the set consists of all non-negative integer solutions of the Diophantine equation
See: Faà di Bruno's formula, Expansions for nearly Gaussian distributions by S. Blinnikov and R. Moessner
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![{\displaystyle y=F(x)G(x)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd363a6a7897619c9e1c29c9d170f30e826d8c4) |
See: General Leibniz rule
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![{\displaystyle y=x^{N}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ac2b53f9712e05ee916a672048bcc185162e28) |
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![{\displaystyle y=[F(x)]^{r}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/354406061c702eebc94b0a45d38966ccf2c342b4) |
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![{\displaystyle y=B^{Ax}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0609c48250271ca08a20fc3f5386880ab74b38fb) |
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For the case of (the exponential function),
the above reduces to:
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![{\displaystyle y=\ln[F(x)]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8020f82005ebdef54cec5c07133481dad5d734b1) |
where is the Kronecker delta.
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![{\displaystyle y=\sin(Ax+B)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b02537d211faf4e8574f002b2781423543df75a5) |
Expanding this by the sine addition formula yields a more clear form to use:
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![{\displaystyle y=\cos(Ax+B)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1b35eeb5f8c5bc7cff1f93cb2e6a24525240cf) |
Expanding by the cosine addition formula:
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![{\displaystyle y=\sinh(Ax+B)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1621db2849491aa6df85597ab898910816893f0e) |
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![{\displaystyle y=\cosh(Ax+B)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2770530776489a7e069ec3cf20ceb26d69d81a66) |
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