Производная (1+log(sin(x)))^n

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Решение

Вы ввели [src]
                 n
(1 + log(sin(x))) 
(log(sin(x))+1)n\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n}
Подробное решение
  1. Заменим u=log(sin(x))+1u = \log{\left (\sin{\left (x \right )} \right )} + 1.

  2. В силу правила, применим: unu^{n} получим nunu\frac{n u^{n}}{u}

  3. Затем примените цепочку правил. Умножим на ddx(log(sin(x))+1)\frac{d}{d x}\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right):

    1. дифференцируем log(sin(x))+1\log{\left (\sin{\left (x \right )} \right )} + 1 почленно:

      1. Производная постоянной 11 равна нулю.

      2. Заменим u=sin(x)u = \sin{\left (x \right )}.

      3. Производная log(u)\log{\left (u \right )} является 1u\frac{1}{u}.

      4. Затем примените цепочку правил. Умножим на ddxsin(x)\frac{d}{d x} \sin{\left (x \right )}:

        1. Производная синуса есть косинус:

          ddxsin(x)=cos(x)\frac{d}{d x} \sin{\left (x \right )} = \cos{\left (x \right )}

        В результате последовательности правил:

        cos(x)sin(x)\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}

      В результате: cos(x)sin(x)\frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}

    В результате последовательности правил:

    n(log(sin(x))+1)ncos(x)(log(sin(x))+1)sin(x)\frac{n \left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n} \cos{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin{\left (x \right )}}

  4. Теперь упростим:

    ntan(x)(log(sin(x))+1)n1\frac{n}{\tan{\left (x \right )}} \left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n - 1}


Ответ:

ntan(x)(log(sin(x))+1)n1\frac{n}{\tan{\left (x \right )}} \left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n - 1}

Первая производная [src]
                   n       
n*(1 + log(sin(x))) *cos(x)
---------------------------
  (1 + log(sin(x)))*sin(x) 
n(log(sin(x))+1)ncos(x)(log(sin(x))+1)sin(x)\frac{n \left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n} \cos{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin{\left (x \right )}}
Вторая производная [src]
                     /        2                  2                            2           \
                   n |     cos (x)            cos (x)                    n*cos (x)        |
n*(1 + log(sin(x))) *|-1 - ------- - ------------------------- + -------------------------|
                     |        2                           2                           2   |
                     \     sin (x)   (1 + log(sin(x)))*sin (x)   (1 + log(sin(x)))*sin (x)/
-------------------------------------------------------------------------------------------
                                      1 + log(sin(x))                                      
n(log(sin(x))+1)nlog(sin(x))+1(ncos2(x)(log(sin(x))+1)sin2(x)1cos2(x)sin2(x)cos2(x)(log(sin(x))+1)sin2(x))\frac{n \left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n}}{\log{\left (\sin{\left (x \right )} \right )} + 1} \left(\frac{n \cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin^{2}{\left (x \right )}} - 1 - \frac{\cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} - \frac{\cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin^{2}{\left (x \right )}}\right)
Третья производная [src]
                     /                                             2                   2                            2                       2    2                            2                           2           \       
                   n |           3                3*n         2*cos (x)           2*cos (x)                    3*cos (x)                   n *cos (x)                  3*n*cos (x)                 3*n*cos (x)        |       
n*(1 + log(sin(x))) *|2 + --------------- - --------------- + --------- + -------------------------- + ------------------------- + -------------------------- - ------------------------- - --------------------------|*cos(x)
                     |    1 + log(sin(x))   1 + log(sin(x))       2                        2    2                           2                       2    2                           2                       2    2   |       
                     \                                         sin (x)    (1 + log(sin(x))) *sin (x)   (1 + log(sin(x)))*sin (x)   (1 + log(sin(x))) *sin (x)   (1 + log(sin(x)))*sin (x)   (1 + log(sin(x))) *sin (x)/       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                   (1 + log(sin(x)))*sin(x)                                                                                                   
n(log(sin(x))+1)ncos(x)(log(sin(x))+1)sin(x)(n2cos2(x)(log(sin(x))+1)2sin2(x)3nlog(sin(x))+13ncos2(x)(log(sin(x))+1)sin2(x)3ncos2(x)(log(sin(x))+1)2sin2(x)+2+2cos2(x)sin2(x)+3log(sin(x))+1+3cos2(x)(log(sin(x))+1)sin2(x)+2cos2(x)(log(sin(x))+1)2sin2(x))\frac{n \left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{n} \cos{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin{\left (x \right )}} \left(\frac{n^{2} \cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{2} \sin^{2}{\left (x \right )}} - \frac{3 n}{\log{\left (\sin{\left (x \right )} \right )} + 1} - \frac{3 n \cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin^{2}{\left (x \right )}} - \frac{3 n \cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{2} \sin^{2}{\left (x \right )}} + 2 + \frac{2 \cos^{2}{\left (x \right )}}{\sin^{2}{\left (x \right )}} + \frac{3}{\log{\left (\sin{\left (x \right )} \right )} + 1} + \frac{3 \cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right) \sin^{2}{\left (x \right )}} + \frac{2 \cos^{2}{\left (x \right )}}{\left(\log{\left (\sin{\left (x \right )} \right )} + 1\right)^{2} \sin^{2}{\left (x \right )}}\right)