Производная 10^(x*tan(x))

Вы ввели

  x*tan(x)
10        

$$10^{x \tan{\left (x \right )}}$$

Подробное решение

1. Заменим $u = x \tan{\left (x \right )}$.

2. $\frac{d}{d u} 10^{u} = 10^{u} \log{\left (10 \right )}$

3. Затем примените цепочку правил. Умножим на $\frac{d}{d x}\left(x \tan{\left (x \right )}\right)$:

   1. Применяем правило производной умножения:

      $\frac{d}{d x}\left(f{\left (x \right )} g{\left (x \right )}\right) = f{\left (x \right )} \frac{d}{d x} g{\left (x \right )} + g{\left (x \right )} \frac{d}{d x} f{\left (x \right )}$

      $f{\left (x \right )} = x$; найдём $\frac{d}{d x} f{\left (x \right )}$:

      1. В силу правила, применим: $x$ получим $1$

      $g{\left (x \right )} = \tan{\left (x \right )}$; найдём $\frac{d}{d x} g{\left (x \right )}$:

      1. Есть несколько способов вычислить эту производную.

         Один из способов:

         1. $\frac{d}{d x} \tan{\left (x \right )} = \frac{1}{\cos^{2}{\left (x \right )}}$

      В результате: $\frac{x}{\cos^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) + \tan{\left (x \right )}$

   В результате последовательности правил:

   $10^{x \tan{\left (x \right )}} \left(\frac{x}{\cos^{2}{\left (x \right )}} \left(\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}\right) + \tan{\left (x \right )}\right) \log{\left (10 \right )}$

4. Теперь упростим:

   $\frac{10^{x \tan{\left (x \right )}}}{\cos^{2}{\left (x \right )}} \left(x + \frac{1}{2} \sin{\left (2 x \right )}\right) \log{\left (10 \right )}$

---

Ответ:

$\frac{10^{x \tan{\left (x \right )}}}{\cos^{2}{\left (x \right )}} \left(x + \frac{1}{2} \sin{\left (2 x \right )}\right) \log{\left (10 \right )}$

Первая производная

  x*tan(x) /  /       2   \         \        
10        *\x*\1 + tan (x)/ + tan(x)/*log(10)

$$10^{x \tan{\left (x \right )}} \left(x \left(\tan^{2}{\left (x \right )} + 1\right) + \tan{\left (x \right )}\right) \log{\left (10 \right )}$$

Вторая производная

           /                                          2                                   \        
  x*tan(x) |         2      /  /       2   \         \                /       2   \       |        
10        *\2 + 2*tan (x) + \x*\1 + tan (x)/ + tan(x)/ *log(10) + 2*x*\1 + tan (x)/*tan(x)/*log(10)

$$10^{x \tan{\left (x \right )}} \left(2 x \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \left(x \left(\tan^{2}{\left (x \right )} + 1\right) + \tan{\left (x \right )}\right)^{2} \log{\left (10 \right )} + 2 \tan^{2}{\left (x \right )} + 2\right) \log{\left (10 \right )}$$

Третья производная

           /                          3                                                                                                                                                    \        
  x*tan(x) |/  /       2   \         \     2         /       2   \ /             /       2   \          2   \     /  /       2   \         \ /       2        /       2   \       \        |        
10        *\\x*\1 + tan (x)/ + tan(x)/ *log (10) + 2*\1 + tan (x)/*\3*tan(x) + x*\1 + tan (x)/ + 2*x*tan (x)/ + 6*\x*\1 + tan (x)/ + tan(x)/*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/*log(10)/*log(10)

$$10^{x \tan{\left (x \right )}} \left(\left(x \left(\tan^{2}{\left (x \right )} + 1\right) + \tan{\left (x \right )}\right)^{3} \log^{2}{\left (10 \right )} + 6 \left(x \left(\tan^{2}{\left (x \right )} + 1\right) + \tan{\left (x \right )}\right) \left(x \left(\tan^{2}{\left (x \right )} + 1\right) \tan{\left (x \right )} + \tan^{2}{\left (x \right )} + 1\right) \log{\left (10 \right )} + 2 \left(\tan^{2}{\left (x \right )} + 1\right) \left(x \left(\tan^{2}{\left (x \right )} + 1\right) + 2 x \tan^{2}{\left (x \right )} + 3 \tan{\left (x \right )}\right)\right) \log{\left (10 \right )}$$

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