Производная 3^(1/(log((5),(x+1))))
Учитель очень удивится увидев твоё верное решение производной 😉
Решение
График
Первая производная
[src]
1
-------------
log(5, x + 1) / d / log(5) \\|
-3 *|-----|---------||| *log(3)
\dxi_2\log(xi_2)//|xi_2=x + 1
-----------------------------------------------------
2
log (5, x + 1)
$$- \frac{3^{\frac{1}{\log{\left (5,x + 1 \right )}}} \log{\left (3 \right )}}{\log^{2}{\left (5,x + 1 \right )}} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }}$$
Вторая производная
[src]
/ 2 2 \
log(1 + x) | 2 / d / log(5) \\| 3 / d / log(5) \\| 4 |
---------- | 1 + ---------- 2*|-----|---------||| *log (1 + x) |-----|---------||| *log (1 + x)*log(3)|
log(5) | log(1 + x) \dxi_2\log(xi_2)//|xi_2=x + 1 \dxi_2\log(xi_2)//|xi_2=x + 1 |
3 *|- -------------- + -------------------------------------------- + -------------------------------------------------|*log(3)
| 2 2 3 |
\ (1 + x) log (5) log (5) /
----------------------------------------------------------------------------------------------------------------------------------------
log(5)
$$\frac{\log{\left (3 \right )}}{\log{\left (5 \right )}} 3^{\frac{\log{\left (x + 1 \right )}}{\log{\left (5 \right )}}} \left(- \frac{1}{\left(x + 1\right)^{2}} \left(1 + \frac{2}{\log{\left (x + 1 \right )}}\right) + \frac{\log^{4}{\left (x + 1 \right )}}{\log^{3}{\left (5 \right )}} \log{\left (3 \right )} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }}^{2} + \frac{2 \log^{3}{\left (x + 1 \right )}}{\log^{2}{\left (5 \right )}} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }}^{2}\right)$$
Третья производная
[src]
/ 3 3 3 \
log(1 + x) | / 2 \ / d / log(5) \\| 4 / 2 \ / d / log(5) \\| 2 6 / d / log(5) \\| 5 / 2 \ / d / log(5) \\| 2 / 2 \ / d / log(5) \\| |
---------- | 2*|1 + ----------| 6*|-----|---------||| *log (1 + x) 2*|1 + ----------| |-----|---------||| *log (3)*log (1 + x) 6*|-----|---------||| *log (1 + x)*log(3) 6*|1 + ----------|*|-----|---------||| *log(1 + x) 3*log (1 + x)*|1 + ----------|*|-----|---------||| *log(3)|
log(5) | 2 \ log(1 + x)/ \dxi_2\log(xi_2)//|xi_2=x + 1 \ log(1 + x)/ \dxi_2\log(xi_2)//|xi_2=x + 1 \dxi_2\log(xi_2)//|xi_2=x + 1 \ log(1 + x)/ \dxi_2\log(xi_2)//|xi_2=1 + x \ log(1 + x)/ \dxi_2\log(xi_2)//|xi_2=1 + x |
3 *|-------------------- + ------------------ - -------------------------------------------- + ------------------- - -------------------------------------------------- - --------------------------------------------------- + ----------------------------------------------------------- + -------------------------------------------------------------------|*log(3)
| 3 2 3 3 3 5 4 2 2 2 |
\(1 + x) *log (1 + x) (1 + x) log (5) (1 + x) *log(1 + x) log (5) log (5) (1 + x) *log(5) (1 + x) *log (5) /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
log(5)
$$\frac{\log{\left (3 \right )}}{\log{\left (5 \right )}} 3^{\frac{\log{\left (x + 1 \right )}}{\log{\left (5 \right )}}} \left(\frac{3 \log{\left (3 \right )} \log^{2}{\left (x + 1 \right )}}{\left(x + 1\right)^{2} \log^{2}{\left (5 \right )}} \left(1 + \frac{2}{\log{\left (x + 1 \right )}}\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }} + \frac{6 \log{\left (x + 1 \right )}}{\left(x + 1\right)^{2} \log{\left (5 \right )}} \left(1 + \frac{2}{\log{\left (x + 1 \right )}}\right) \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }} + \frac{1}{\left(x + 1\right)^{3}} \left(2 + \frac{4}{\log{\left (x + 1 \right )}}\right) + \frac{2 + \frac{4}{\log{\left (x + 1 \right )}}}{\left(x + 1\right)^{3} \log{\left (x + 1 \right )}} - \frac{\log^{2}{\left (3 \right )}}{\log^{5}{\left (5 \right )}} \log^{6}{\left (x + 1 \right )} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }}^{3} - \frac{6 \log^{5}{\left (x + 1 \right )}}{\log^{4}{\left (5 \right )}} \log{\left (3 \right )} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }}^{3} - \frac{6 \log^{4}{\left (x + 1 \right )}}{\log^{3}{\left (5 \right )}} \left. \frac{d}{d \xi_{2}}\left(\frac{\log{\left (5 \right )}}{\log{\left (\xi_{2} \right )}}\right) \right|_{\substack{ \xi_{2}=x + 1 }}^{3} + \frac{2}{\left(x + 1\right)^{3} \log^{2}{\left (x + 1 \right )}}\right)$$