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Derivative of tanh(x)
Derivative of tanx
Derivative of tg2x
Derivative of e^(sin(1-3*x)^(2))
Identical expressions
logsin5x* nine ^(tg13x^ six)
logarithm of sinus of 5x multiply by 9 to the power of (tg13x to the power of 6)
logarithm of sinus of 5x multiply by nine to the power of (tg13x to the power of six)
logsin5x*9(tg13x6)
logsin5x*9tg13x6
logsin5x*9^(tg13x⁶)
logsin5x9^(tg13x^6)
logsin5x9(tg13x6)
logsin5x9tg13x6
logsin5x9^tg13x^6

The derivative of the function/ logsin5x*9^(tg13x^6)

Derivative of logsin5x*9^(tg13x^6)

The solution

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                  6      
               tan (13*x)
log(sin(5*x))*9

$$9^{\tan^{6}{\left(13 x \right)}} \log{\left(\sin{\left(5 x \right)} \right)}$$

log(sin(5*x))*9^(tan(13*x)^6)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(\sin{\left(5 x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(5 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $5$
    The result of the chain rule is:
    
    $5 \cos{\left(5 x \right)}$
  The result of the chain rule is:
  
  $\frac{5 \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$
$g{\left(x \right)} = 9^{\tan^{6}{\left(13 x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \tan^{6}{\left(13 x \right)}$ .
2. $\frac{d}{d u} 9^{u} = 9^{u} \log{\left(9 \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan^{6}{\left(13 x \right)}$ :
  1. Let $u = \tan{\left(13 x \right)}$ .
  2. Apply the power rule: $u^{6}$ goes to $6 u^{5}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(13 x \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(13 x \right)} = \frac{\sin{\left(13 x \right)}}{\cos{\left(13 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(13 x \right)}$ and $g{\left(x \right)} = \cos{\left(13 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Let $u = 13 x$ .
      2. The derivative of sine is cosine:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 13 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $13$
        
        The result of the chain rule is:
        
        $13 \cos{\left(13 x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = 13 x$ .
      2. The derivative of cosine is negative sine:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 13 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $13$
        
        The result of the chain rule is:
        
        $- 13 \sin{\left(13 x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{13 \sin^{2}{\left(13 x \right)} + 13 \cos^{2}{\left(13 x \right)}}{\cos^{2}{\left(13 x \right)}}$
    The result of the chain rule is:
    
    $\frac{6 \left(13 \sin^{2}{\left(13 x \right)} + 13 \cos^{2}{\left(13 x \right)}\right) \tan^{5}{\left(13 x \right)}}{\cos^{2}{\left(13 x \right)}}$
  The result of the chain rule is:
  
  $\frac{6 \cdot 9^{\tan^{6}{\left(13 x \right)}} \left(13 \sin^{2}{\left(13 x \right)} + 13 \cos^{2}{\left(13 x \right)}\right) \log{\left(9 \right)} \tan^{5}{\left(13 x \right)}}{\cos^{2}{\left(13 x \right)}}$
The result is: $\frac{6 \cdot 9^{\tan^{6}{\left(13 x \right)}} \left(13 \sin^{2}{\left(13 x \right)} + 13 \cos^{2}{\left(13 x \right)}\right) \log{\left(9 \right)} \log{\left(\sin{\left(5 x \right)} \right)} \tan^{5}{\left(13 x \right)}}{\cos^{2}{\left(13 x \right)}} + \frac{5 \cdot 9^{\tan^{6}{\left(13 x \right)}} \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$
Now simplify:

$\frac{156 \cdot 3^{2 \tan^{6}{\left(13 x \right)}} \log{\left(3 \right)} \log{\left(\sin{\left(5 x \right)} \right)} \sin^{5}{\left(13 x \right)}}{\cos^{7}{\left(13 x \right)}} + \frac{5 \cdot 9^{\tan^{6}{\left(13 x \right)}}}{\tan{\left(5 x \right)}}$

The answer is:

$\frac{156 \cdot 3^{2 \tan^{6}{\left(13 x \right)}} \log{\left(3 \right)} \log{\left(\sin{\left(5 x \right)} \right)} \sin^{5}{\left(13 x \right)}}{\cos^{7}{\left(13 x \right)}} + \frac{5 \cdot 9^{\tan^{6}{\left(13 x \right)}}}{\tan{\left(5 x \right)}}$

The first derivative [src]

      6                                                                                  
   tan (13*x)                6                                                           
5*9          *cos(5*x)    tan (13*x)    5       /           2      \                     
---------------------- + 9          *tan (13*x)*\78 + 78*tan (13*x)/*log(9)*log(sin(5*x))
       sin(5*x)

$$9^{\tan^{6}{\left(13 x \right)}} \left(78 \tan^{2}{\left(13 x \right)} + 78\right) \log{\left(9 \right)} \log{\left(\sin{\left(5 x \right)} \right)} \tan^{5}{\left(13 x \right)} + \frac{5 \cdot 9^{\tan^{6}{\left(13 x \right)}} \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}}$$

Simplify

The second derivative [src]

    6       /            2               5       /       2      \                                                                                                                                  \
 tan (13*x) |      25*cos (5*x)   780*tan (13*x)*\1 + tan (13*x)/*cos(5*x)*log(9)           4       /       2      \ /         2              6       /       2      \       \                     |
9          *|-25 - ------------ + ----------------------------------------------- + 1014*tan (13*x)*\1 + tan (13*x)/*\5 + 7*tan (13*x) + 6*tan (13*x)*\1 + tan (13*x)/*log(9)/*log(9)*log(sin(5*x))|
            |          2                              sin(5*x)                                                                                                                                     |
            \       sin (5*x)                                                                                                                                                                      /

$$9^{\tan^{6}{\left(13 x \right)}} \left(1014 \left(\tan^{2}{\left(13 x \right)} + 1\right) \left(6 \left(\tan^{2}{\left(13 x \right)} + 1\right) \log{\left(9 \right)} \tan^{6}{\left(13 x \right)} + 7 \tan^{2}{\left(13 x \right)} + 5\right) \log{\left(9 \right)} \log{\left(\sin{\left(5 x \right)} \right)} \tan^{4}{\left(13 x \right)} + \frac{780 \left(\tan^{2}{\left(13 x \right)} + 1\right) \log{\left(9 \right)} \cos{\left(5 x \right)} \tan^{5}{\left(13 x \right)}}{\sin{\left(5 x \right)}} - 25 - \frac{25 \cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right)$$

Simplify

The third derivative [src]

              /    /       2     \                                                                                                                                                                                                                                                                                                                                                                                                                                            \
              |    |    cos (5*x)|                                                                                                                                                                                                                                                                                                                                                                                                                                            |
              |125*|1 + ---------|*cos(5*x)                                                                                                                                                                                                                                                                                                                                                                                                                                   |
      6       |    |       2     |                                             /       2     \                                            /                                  2                                                       2                                                                                  2                  \                                4       /       2      \ /         2              6       /       2      \       \                |
   tan (13*x) |    \    sin (5*x)/                    5       /       2      \ |    cos (5*x)|                   3       /       2      \ |     4            /       2      \          2       /       2      \      /       2      \     2       12               8       /       2      \             /       2      \     6             |                        7605*tan (13*x)*\1 + tan (13*x)/*\5 + 7*tan (13*x) + 6*tan (13*x)*\1 + tan (13*x)/*log(9)/*cos(5*x)*log(9)|
2*9          *|---------------------------- - 2925*tan (13*x)*\1 + tan (13*x)/*|1 + ---------|*log(9) + 13182*tan (13*x)*\1 + tan (13*x)/*\2*tan (13*x) + 10*\1 + tan (13*x)/  + 16*tan (13*x)*\1 + tan (13*x)/ + 18*\1 + tan (13*x)/ *log (9)*tan  (13*x) + 18*tan (13*x)*\1 + tan (13*x)/*log(9) + 45*\1 + tan (13*x)/ *tan (13*x)*log(9)/*log(9)*log(sin(5*x)) + ----------------------------------------------------------------------------------------------------------|
              |          sin(5*x)                                              |       2     |                                                                                                                                                                                                                                                                                                                       sin(5*x)                                                 |
              \                                                                \    sin (5*x)/                                                                                                                                                                                                                                                                                                                                                                                /

$$2 \cdot 9^{\tan^{6}{\left(13 x \right)}} \left(- 2925 \left(1 + \frac{\cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \left(\tan^{2}{\left(13 x \right)} + 1\right) \log{\left(9 \right)} \tan^{5}{\left(13 x \right)} + \frac{125 \left(1 + \frac{\cos^{2}{\left(5 x \right)}}{\sin^{2}{\left(5 x \right)}}\right) \cos{\left(5 x \right)}}{\sin{\left(5 x \right)}} + \frac{7605 \left(\tan^{2}{\left(13 x \right)} + 1\right) \left(6 \left(\tan^{2}{\left(13 x \right)} + 1\right) \log{\left(9 \right)} \tan^{6}{\left(13 x \right)} + 7 \tan^{2}{\left(13 x \right)} + 5\right) \log{\left(9 \right)} \cos{\left(5 x \right)} \tan^{4}{\left(13 x \right)}}{\sin{\left(5 x \right)}} + 13182 \left(\tan^{2}{\left(13 x \right)} + 1\right) \left(18 \left(\tan^{2}{\left(13 x \right)} + 1\right)^{2} \log{\left(9 \right)}^{2} \tan^{12}{\left(13 x \right)} + 45 \left(\tan^{2}{\left(13 x \right)} + 1\right)^{2} \log{\left(9 \right)} \tan^{6}{\left(13 x \right)} + 10 \left(\tan^{2}{\left(13 x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(13 x \right)} + 1\right) \log{\left(9 \right)} \tan^{8}{\left(13 x \right)} + 16 \left(\tan^{2}{\left(13 x \right)} + 1\right) \tan^{2}{\left(13 x \right)} + 2 \tan^{4}{\left(13 x \right)}\right) \log{\left(9 \right)} \log{\left(\sin{\left(5 x \right)} \right)} \tan^{3}{\left(13 x \right)}\right)$$

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Derivative of logsin5x*9^(tg13x^6)

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