# Đạo hàm hàm số đặc biệt

 Đạo hàm hàm số đặc biệt Đạo hàm hàm số $\Gamma (x)$ $\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt$ $\Gamma (x)$ $\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)=\Gamma (x)\psi (x)$ $\zeta (x)$ $-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \!$ $\zeta (x)$ $-\sum _{p{\text{ prime}}}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2}}}\prod _{q{\text{ prime}},q\neq p}{\frac {1}{1-q^{-x}}}\!$ Hàm số Đạo hàm bậc N $y=F(G(x))\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=n!\displaystyle \sum _{\{k_{m}\}}^{}{\dfrac {\mathrm {d} ^{r}}{\mathrm {d} z^{r}}}F(z)|_{z=G(x)}\displaystyle \prod _{m=1}^{n}{\dfrac {1}{k_{m}!}}\left({\dfrac {1}{m!}}{\dfrac {\mathrm {d} ^{m}}{\mathrm {d} x^{m}}}G(x)\right)^{k_{m}}\!$ where $r=\displaystyle \sum _{m=1}^{n}k_{m}\!$ and the set $\{k_{m}\}\!$ consists of all non-negative integer solutions of the Diophantine equation $\displaystyle \sum _{m=1}^{n}mk_{m}=n\!$ See: Faà di Bruno's formula, Expansions for nearly Gaussian distributions by S. Blinnikov and R. Moessner $y=F(x)G(x)\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=\displaystyle \sum _{k=0}^{n}\displaystyle {\binom {n}{k}}{\dfrac {\mathrm {d} ^{n-k}}{\mathrm {d} x^{n-k}}}F(x){\dfrac {\mathrm {d} ^{k}}{\mathrm {d} x^{k}}}G(x)\!$ $y=x^{N}\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=\displaystyle \prod _{r=1}^{n}(N-r+1)x^{N-n}\!$ $y=[F(x)]^{r}\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=r\displaystyle {\binom {n-r}{n}}\displaystyle \sum _{j=0}^{n}{\dfrac {(-1)^{j}}{r-j}}{\binom {n}{j}}[F(x)]^{r-j}{\dfrac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}[F(x)]^{j}}\!$ $y=B^{Ax}\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}B^{Ax}\left(\ln {B}\right)^{n}\!$ For the case of $B=\exp(1)=e\!$ (the exponential function), the above reduces to: $y=e^{Ax}\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}e^{Ax}\!$ $y=\ln[F(x)]\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=\delta _{n}\ln[F(x)]+\displaystyle \sum _{j=1}^{n}{\dfrac {(-1)^{j-1}}{j}}{\binom {n}{j}}{\dfrac {1}{[F(x)]^{j}}}{\dfrac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}[F(x)]^{j}\!$ where $\delta _{n}={\begin{cases}1&n=0\\0&n\neq 0\\\end{cases}}\!$ is the Kronecker delta. $y=\sin(Ax+B)\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\sin \left(Ax+B+{\frac {n\pi }{2}}\right)\!$ Expanding this by the sine addition formula yields a more clear form to use: ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\left[\cos \left({\dfrac {n\pi }{2}}\right)\sin \left(Ax+B\right)+\sin \left({\dfrac {n\pi }{2}}\right)\cos \left(Ax+B\right)\right]\!$ $y=\cos(Ax+B)\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\cos \left(Ax+B+{\frac {n\pi }{2}}\right)\!$ Expanding by the cosine addition formula: ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=A^{n}\left[\cos \left({\dfrac {n\pi }{2}}\right)\cos \left(Ax+B\right)-\sin \left({\dfrac {n\pi }{2}}\right)\sin \left(Ax+B\right)\right]\!$ $y=\sinh(Ax+B)\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=(-iA^{n})\sinh \left(Ax+B+{\dfrac {in\pi }{2}}\right)\!$ $y=\cosh(Ax+B)\!$ ${\dfrac {\mathrm {d} ^{n}y}{\mathrm {d} x^{n}}}=(\pm iA^{n})\cosh \left(Ax+B\mp {\dfrac {in\pi }{2}}\right)\!$ 