Производная ((x+1)/(x-1))^(1/5)*log(3,x^(2)+x+4)
Решение
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_______ / x + 1 / 2 \ 5 / ----- *log\3, x + x + 4/ \/ x - 1
$$\sqrt[5]{\frac{x + 1}{x - 1}} \log{\left (3,x^{2} + x + 4 \right )}$$
Первая производная
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_______
/ x + 1 / 1 x + 1 \ / 2 \
5 / ----- *(x - 1)*|--------- - ----------|*log\3, x + x + 4/
_______ \/ x - 1 |5*(x - 1) 2|
/ x + 1 / d / log(3) \\| \ 5*(x - 1) /
5 / ----- *(1 + 2*x)*|-----|---------||| 2 + ---------------------------------------------------------------
\/ x - 1 \dxi_2\log(xi_2)//|xi_2=x + x + 4 x + 1
$$\frac{\sqrt[5]{\frac{x + 1}{x - 1}}}{x + 1} \left(x - 1\right) \left(\frac{1}{5 x - 5} - \frac{x + 1}{5 \left(x - 1\right)^{2}}\right) \log{\left (3,x^{2} + x + 4 \right )} + \sqrt[5]{\frac{x + 1}{x - 1}} \left(2 x + 1\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }}$$
Вторая производная
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/ 2 2 / 2 \ \
| / 1 + x \ / 1 + x \ / 1 + x \ / d / log(3) \\| (1 + 2*x) *|1 + ---------------|*log(3) / 1 + x \ |
________ | |1 - ------|*log(3) |1 - ------| *log(3) 2*(1 + 2*x)*|1 - ------|*|-----|---------||| 2 | / 2\| |1 - ------|*log(3) |
/ 1 + x | / d / log(3) \\| \ -1 + x/ \ -1 + x/ \ -1 + x/ \dxi_2\log(xi_2)//|xi_2=4 + x + x \ log\4 + x + x // \ -1 + x/ |
5 / ------ *|2*|-----|---------||| 2 - -------------------------- + --------------------------- + ----------------------------------------------------------- + --------------------------------------- - ----------------------------------|
\/ -1 + x | \dxi_2\log(xi_2)//|xi_2=x + x + 4 2 / 2\ 2 / 2\ 5*(1 + x) 2 / 2\|
| 5*(1 + x) *log\4 + x + x / 25*(1 + x) *log\4 + x + x / / 2\ 2/ 2\ 5*(1 + x)*(-1 + x)*log\4 + x + x /|
\ \4 + x + x / *log \4 + x + x / /
$$\sqrt[5]{\frac{x + 1}{x - 1}} \left(\frac{\left(1 - \frac{x + 1}{x - 1}\right)^{2} \log{\left (3 \right )}}{25 \left(x + 1\right)^{2} \log{\left (x^{2} + x + 4 \right )}} + \frac{2}{5 x + 5} \left(1 - \frac{x + 1}{x - 1}\right) \left(2 x + 1\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }} - \frac{\left(1 - \frac{x + 1}{x - 1}\right) \log{\left (3 \right )}}{5 \left(x + 1\right)^{2} \log{\left (x^{2} + x + 4 \right )}} - \frac{\left(1 - \frac{x + 1}{x - 1}\right) \log{\left (3 \right )}}{5 \left(x - 1\right) \left(x + 1\right) \log{\left (x^{2} + x + 4 \right )}} + \frac{\left(1 + \frac{2}{\log{\left (x^{2} + x + 4 \right )}}\right) \left(2 x + 1\right)^{2} \log{\left (3 \right )}}{\left(x^{2} + x + 4\right)^{2} \log^{2}{\left (x^{2} + x + 4 \right )}} + 2 \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }}\right)$$
Третья производная
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/ 2 3 2 3 / 2 \ 3 / 2 \ / 2 \ 2 2 / 2 \ / 1 + x \ \
| / 1 + x \ / d / log(3) \\| / 1 + x \ / d / log(3) \\| / 1 + x \ / 1 + x \ / 1 + x \ / 1 + x \ / d / log(3) \\| 2*(1 + 2*x) *|1 + ---------------|*log(3) 2*(1 + 2*x) *|1 + ---------------|*log(3) 6*(1 + 2*x)*|1 + ---------------|*log(3) / 1 + x \ / d / log(3) \\| / 1 + x \ / 1 + x \ / 1 + x \ 3*(1 + 2*x) *|1 + ---------------|*|1 - ------|*log(3)|
________ |6*|1 - ------|*|-----|---------||| 2 3 3*(1 + 2*x)*|1 - ------|*|-----|---------||| 2 3*|1 - ------| *log(3) |1 - ------| *log(3) 2*|1 - ------|*log(3) 3*|1 - ------| *(1 + 2*x)*|-----|---------||| 2 | / 2\| | / 2\| | / 2\| 3*(1 + 2*x)*|1 - ------|*|-----|---------||| 2 3*|1 - ------| *log(3) 2*|1 - ------|*log(3) 2*|1 - ------|*log(3) | / 2\| \ -1 + x/ |
/ 1 + x | \ -1 + x/ \dxi_2\log(xi_2)//|xi_2=4 + x + x 2*(1 + 2*x) *log(3) \ -1 + x/ \dxi_2\log(xi_2)//|xi_2=4 + x + x \ -1 + x/ \ -1 + x/ \ -1 + x/ \ -1 + x/ \dxi_2\log(xi_2)//|xi_2=4 + x + x \ log\4 + x + x // \ log\4 + x + x // \ log\4 + x + x // \ -1 + x/ \dxi_2\log(xi_2)//|xi_2=4 + x + x \ -1 + x/ \ -1 + x/ \ -1 + x/ \ log\4 + x + x // |
5 / ------ *|------------------------------------------------- - ------------------------------ - ----------------------------------------------------------- - --------------------------- + ---------------------------- + -------------------------- + ------------------------------------------------------------ - ----------------------------------------- - ----------------------------------------- + ---------------------------------------- - ----------------------------------------------------------- - ------------------------------------ + ----------------------------------- + ----------------------------------- + ------------------------------------------------------|
\/ -1 + x | 5*(1 + x) 3 2 3 / 2\ 3 / 2\ 3 / 2\ 2 3 3 2 5*(1 + x)*(-1 + x) 2 / 2\ 2 / 2\ 2 / 2\ 2 |
| / 2\ 4/ 2\ 5*(1 + x) 25*(1 + x) *log\4 + x + x / 125*(1 + x) *log\4 + x + x / 5*(1 + x) *log\4 + x + x / 25*(1 + x) / 2\ 3/ 2\ / 2\ 2/ 2\ / 2\ 2/ 2\ 25*(1 + x) *(-1 + x)*log\4 + x + x / 5*(1 + x)*(-1 + x) *log\4 + x + x / 5*(1 + x) *(-1 + x)*log\4 + x + x / / 2\ 2/ 2\ |
\ \4 + x + x / *log \4 + x + x / \4 + x + x / *log \4 + x + x / \4 + x + x / *log \4 + x + x / \4 + x + x / *log \4 + x + x / 5*(1 + x)*\4 + x + x / *log \4 + x + x / /
$$\sqrt[5]{\frac{x + 1}{x - 1}} \left(\frac{\left(1 - \frac{x + 1}{x - 1}\right)^{3} \log{\left (3 \right )}}{125 \left(x + 1\right)^{3} \log{\left (x^{2} + x + 4 \right )}} + \frac{3 \left(1 - \frac{x + 1}{x - 1}\right)^{2}}{25 \left(x + 1\right)^{2}} \left(2 x + 1\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }} - \frac{3 \left(1 - \frac{x + 1}{x - 1}\right)^{2} \log{\left (3 \right )}}{25 \left(x + 1\right)^{3} \log{\left (x^{2} + x + 4 \right )}} - \frac{3 \left(1 - \frac{x + 1}{x - 1}\right)^{2} \log{\left (3 \right )}}{25 \left(x - 1\right) \left(x + 1\right)^{2} \log{\left (x^{2} + x + 4 \right )}} + \frac{3 \left(1 - \frac{x + 1}{x - 1}\right) \left(1 + \frac{2}{\log{\left (x^{2} + x + 4 \right )}}\right) \left(2 x + 1\right)^{2} \log{\left (3 \right )}}{5 \left(x + 1\right) \left(x^{2} + x + 4\right)^{2} \log^{2}{\left (x^{2} + x + 4 \right )}} + \frac{6}{5 x + 5} \left(1 - \frac{x + 1}{x - 1}\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }} - \frac{3}{5 \left(x + 1\right)^{2}} \left(1 - \frac{x + 1}{x - 1}\right) \left(2 x + 1\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }} + \frac{2 \left(1 - \frac{x + 1}{x - 1}\right) \log{\left (3 \right )}}{5 \left(x + 1\right)^{3} \log{\left (x^{2} + x + 4 \right )}} - \frac{3}{5 \left(x - 1\right) \left(x + 1\right)} \left(1 - \frac{x + 1}{x - 1}\right) \left(2 x + 1\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (3 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x^{2} + x + 4 }} + \frac{2 \left(1 - \frac{x + 1}{x - 1}\right) \log{\left (3 \right )}}{5 \left(x - 1\right) \left(x + 1\right)^{2} \log{\left (x^{2} + x + 4 \right )}} + \frac{2 \left(1 - \frac{x + 1}{x - 1}\right) \log{\left (3 \right )}}{5 \left(x - 1\right)^{2} \left(x + 1\right) \log{\left (x^{2} + x + 4 \right )}} - \frac{2 \left(1 + \frac{2}{\log{\left (x^{2} + x + 4 \right )}}\right) \left(2 x + 1\right)^{3} \log{\left (3 \right )}}{\left(x^{2} + x + 4\right)^{3} \log^{2}{\left (x^{2} + x + 4 \right )}} - \frac{2 \left(1 + \frac{2}{\log{\left (x^{2} + x + 4 \right )}}\right) \left(2 x + 1\right)^{3} \log{\left (3 \right )}}{\left(x^{2} + x + 4\right)^{3} \log^{3}{\left (x^{2} + x + 4 \right )}} + \frac{6 \left(1 + \frac{2}{\log{\left (x^{2} + x + 4 \right )}}\right) \left(2 x + 1\right) \log{\left (3 \right )}}{\left(x^{2} + x + 4\right)^{2} \log^{2}{\left (x^{2} + x + 4 \right )}} - \frac{2 \left(2 x + 1\right)^{3} \log{\left (3 \right )}}{\left(x^{2} + x + 4\right)^{3} \log^{4}{\left (x^{2} + x + 4 \right )}}\right)$$