Производная 3^(1/(log((5),(x+1))))

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Решение

Вы ввели [src]
       1      
 -------------
 log(5, x + 1)
3             
31log(5,x+1)3^{\frac{1}{\log{\left (5,x + 1 \right )}}}
График
02468-8-6-4-2-1010010
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)                   
31log(5,x+1)log(3)log2(5,x+1)ddξ2(log(5)log(ξ2))ξ2=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)                                                                 
log(3)log(5)3log(x+1)log(5)(1(x+1)2(1+2log(x+1))+log4(x+1)log3(5)log(3)ddξ2(log(5)log(ξ2))ξ2=x+12+2log3(x+1)log2(5)ddξ2(log(5)log(ξ2))ξ2=x+12)\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)                         /       
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                                                                                                                                                                                      log(5)                                                                                                                                                                                      
log(3)log(5)3log(x+1)log(5)(3log(3)log2(x+1)(x+1)2log2(5)(1+2log(x+1))ddξ2(log(5)log(ξ2))ξ2=x+1+6log(x+1)(x+1)2log(5)(1+2log(x+1))ddξ2(log(5)log(ξ2))ξ2=x+1+1(x+1)3(2+4log(x+1))+2+4log(x+1)(x+1)3log(x+1)log2(3)log5(5)log6(x+1)ddξ2(log(5)log(ξ2))ξ2=x+136log5(x+1)log4(5)log(3)ddξ2(log(5)log(ξ2))ξ2=x+136log4(x+1)log3(5)ddξ2(log(5)log(ξ2))ξ2=x+13+2(x+1)3log2(x+1))\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)