Производная sin(cos(tan(sec(log(x)))))

Учитель очень удивится увидев твоё верное решение производной 😼

Решение

Вы ввели [src]

sin(cos(tan(sec(log(x)))))

$$\sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )}$$

Подробное решение

Заменим $u = \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )}$ .
Производная синуса есть косинус:
$\frac{d}{d u} \sin{\left (u \right )} = \cos{\left (u \right )}$
Затем примените цепочку правил. Умножим на $\frac{d}{d x} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )}$ :
1. Заменим $u = \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}$ .
2. Производная косинус есть минус синус:
  $\frac{d}{d u} \cos{\left (u \right )} = - \sin{\left (u \right )}$
3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}$ :
  1. Есть несколько способов вычислить эту производную.
    Один из способов:
    1. Заменим $u = \sec{\left (\log{\left (x \right )} \right )}$ .
    2. $\frac{d}{d u} \tan{\left (u \right )} = \frac{1}{\cos^{2}{\left (u \right )}}$
    3. Затем примените цепочку правил. Умножим на $\frac{d}{d x} \sec{\left (\log{\left (x \right )} \right )}$ :
      1. Есть несколько способов вычислить эту производную.
        Один из способов:
        Заменим $u = \log{\left (x \right )}$ .
        Производная секанса есть секанс, умноженный на тангенс:
        $\frac{d}{d u} \sec{\left (u \right )} = \tan{\left (u \right )} \sec{\left (u \right )}$
        Затем примените цепочку правил. Умножим на $\frac{d}{d x} \log{\left (x \right )}$ :
        Производная $\log{\left (x \right )}$ является $\frac{1}{x}$ .
        В результате последовательности правил:
        $\frac{1}{x} \tan{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )}$
      В результате последовательности правил:
      $\frac{\sin{\left (\log{\left (x \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )} \cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}$
  В результате последовательности правил:
  $- \frac{\sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )}}{\cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}} \left(\frac{\sin^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )}} \sin{\left (\log{\left (x \right )} \right )} + \frac{\cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )}} \sin{\left (\log{\left (x \right )} \right )}\right)$
В результате последовательности правил:
$- \frac{\cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )}}{\cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}} \left(\frac{\sin^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )}} \sin{\left (\log{\left (x \right )} \right )} + \frac{\cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )}} \sin{\left (\log{\left (x \right )} \right )}\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )}$
Теперь упростим:
$- \frac{\sin{\left (\log{\left (x \right )} \right )} \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )} \cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}$

Ответ:

$- \frac{\sin{\left (\log{\left (x \right )} \right )} \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )}}{x \cos^{2}{\left (\log{\left (x \right )} \right )} \cos^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )}}$

График

Первая производная [src]

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-\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*sec(log(x))*sin(tan(sec(log(x))))*tan(log(x)) 
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                                                x

$$- \frac{1}{x} \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )}$$

Упростить

Вторая производная [src]

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\1 + tan (sec(log(x)))/*\cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x))))*tan(log(x)) - tan (log(x))*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - \1 + tan (log(x))/*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - sin (tan(sec(log(x))))*tan (log(x))*\1 + tan (sec(log(x)))/*sec(log(x))*sin(cos(tan(sec(log(x))))) - tan (log(x))*\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*cos(tan(sec(log(x))))*sec(log(x)) - 2*tan (log(x))*cos(cos(tan(sec(log(x)))))*sec(log(x))*sin(tan(sec(log(x))))*tan(sec(log(x)))/*sec(log(x))
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                                                                                                                                                                                                                                                                       x

$$\frac{1}{x^{2}} \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \left(- \left(\tan^{2}{\left (\log{\left (x \right )} \right )} + 1\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} - \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \sin^{2}{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - 2 \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} + \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )}\right) \sec{\left (\log{\left (x \right )} \right )}$$

Упростить

Третья производная [src]

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\1 + tan (sec(log(x)))/*\- tan (log(x))*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - 2*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x))))*tan(log(x)) + 3*tan (log(x))*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) + 3*\1 + tan (log(x))/*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - 5*\1 + tan (log(x))/*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x))))*tan(log(x)) + \1 + tan (sec(log(x)))/ *sec (log(x))*sin (tan(sec(log(x))))*tan (log(x))*cos(cos(tan(sec(log(x))))) + \1 + tan (sec(log(x)))/ *sec (log(x))*tan (log(x))*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - 6*tan (log(x))*cos(cos(tan(sec(log(x)))))*sec(log(x))*sin(tan(sec(log(x))))*tan(sec(log(x))) - 4*sec (log(x))*tan (log(x))*tan (sec(log(x)))*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - 3*sin (tan(sec(log(x))))*tan (log(x))*\1 + tan (sec(log(x)))/*sec(log(x))*sin(cos(tan(sec(log(x))))) - 3*tan (log(x))*\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*cos(tan(sec(log(x))))*sec(log(x)) - 2*sec (log(x))*tan (log(x))*\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) + 3*sin (tan(sec(log(x))))*tan (log(x))*\1 + tan (sec(log(x)))/*sec(log(x))*sin(cos(tan(sec(log(x))))) + 3*tan (log(x))*\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*cos(tan(sec(log(x))))*sec(log(x)) + 6*tan (log(x))*cos(cos(tan(sec(log(x)))))*sec(log(x))*sin(tan(sec(log(x))))*tan(sec(log(x))) - 6*sec (log(x))*sin (tan(sec(log(x))))*tan (log(x))*\1 + tan (sec(log(x)))/*sin(cos(tan(sec(log(x)))))*tan(sec(log(x))) - 6*sec (log(x))*tan (log(x))*\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*cos(tan(sec(log(x))))*tan(sec(log(x))) - 6*\1 + tan (log(x))/*cos(cos(tan(sec(log(x)))))*sec(log(x))*sin(tan(sec(log(x))))*tan(log(x))*tan(sec(log(x))) - 3*\1 + tan (sec(log(x)))/ *sec (log(x))*tan (log(x))*cos(tan(sec(log(x))))*sin(cos(tan(sec(log(x)))))*sin(tan(sec(log(x)))) - 3*sin (tan(sec(log(x))))*\1 + tan (log(x))/*\1 + tan (sec(log(x)))/*sec(log(x))*sin(cos(tan(sec(log(x)))))*tan(log(x)) - 3*\1 + tan (log(x))/*\1 + tan (sec(log(x)))/*cos(cos(tan(sec(log(x)))))*cos(tan(sec(log(x))))*sec(log(x))*tan(log(x))/*sec(log(x))
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$$\frac{1}{x^{3}} \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \left(- 3 \left(\tan^{2}{\left (\log{\left (x \right )} \right )} + 1\right) \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \sin^{2}{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - 3 \left(\tan^{2}{\left (\log{\left (x \right )} \right )} + 1\right) \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - 6 \left(\tan^{2}{\left (\log{\left (x \right )} \right )} + 1\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - 5 \left(\tan^{2}{\left (\log{\left (x \right )} \right )} + 1\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )} + 3 \left(\tan^{2}{\left (\log{\left (x \right )} \right )} + 1\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} - 3 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right)^{2} \sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} + \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right)^{2} \sin^{3}{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} + \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right)^{2} \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} - 6 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \sin^{2}{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} - 3 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \sin^{2}{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} + 3 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \sin^{2}{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - 2 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} - 6 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} - 3 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} + 3 \left(\tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} + 1\right) \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - 4 \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \tan^{2}{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec^{2}{\left (\log{\left (x \right )} \right )} - 6 \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} - \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{3}{\left (\log{\left (x \right )} \right )} + 6 \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} \tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \sec{\left (\log{\left (x \right )} \right )} + 3 \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan^{2}{\left (\log{\left (x \right )} \right )} - 2 \sin{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \cos{\left (\cos{\left (\tan{\left (\sec{\left (\log{\left (x \right )} \right )} \right )} \right )} \right )} \tan{\left (\log{\left (x \right )} \right )}\right) \sec{\left (\log{\left (x \right )} \right )}$$

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Производная sin(cos(tan(sec(log(x)))))

Учитель очень удивится увидев твоё верное решение производной 😼

График:

Кусочно-заданная:

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