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Identical expressions
y=tg^ three (7x)/log(3x+ two)
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Similar expressions
y=tg^3(7x)/log(3x-2)

The derivative of the function/ y=tg^3(7x)/log(3x+2)

Derivative of y=tg^3(7x)/log(3x+2)

The solution

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    3       
 tan (7*x)  
------------
log(3*x + 2)

$$\frac{\tan^{3}{\left(7 x \right)}}{\log{\left(3 x + 2 \right)}}$$

tan(7*x)^3/log(3*x + 2)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \tan^{3}{\left(7 x \right)}$ and $g{\left(x \right)} = \log{\left(3 x + 2 \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \tan{\left(7 x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(7 x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(7 x \right)} = \frac{\sin{\left(7 x \right)}}{\cos{\left(7 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(7 x \right)}$ and $g{\left(x \right)} = \cos{\left(7 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 7 x$ .
    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 $\frac{d}{d x} 7 x$ :
      1. 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: $7$
      The result of the chain rule is:
      
      $7 \cos{\left(7 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 7 x$ .
    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 $\frac{d}{d x} 7 x$ :
      1. 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: $7$
      The result of the chain rule is:
      
      $- 7 \sin{\left(7 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}}$
  The result of the chain rule is:
  
  $\frac{3 \left(7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}\right) \tan^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x + 2$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 x + 2\right)$ :
  1. Differentiate $3 x + 2$ term by term:
    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: $3$
    2. The derivative of the constant $2$ is zero.
    The result is: $3$
  The result of the chain rule is:
  
  $\frac{3}{3 x + 2}$
Now plug in to the quotient rule:

$\frac{\frac{3 \left(7 \sin^{2}{\left(7 x \right)} + 7 \cos^{2}{\left(7 x \right)}\right) \log{\left(3 x + 2 \right)} \tan^{2}{\left(7 x \right)}}{\cos^{2}{\left(7 x \right)}} - \frac{3 \tan^{3}{\left(7 x \right)}}{3 x + 2}}{\log{\left(3 x + 2 \right)}^{2}}$
Now simplify:

$\frac{3 \left(7 \left(3 x + 2\right) \log{\left(3 x + 2 \right)} - \frac{\sin{\left(14 x \right)}}{2}\right) \tan^{2}{\left(7 x \right)}}{\left(3 x + 2\right) \log{\left(3 x + 2 \right)}^{2} \cos^{2}{\left(7 x \right)}}$

The answer is:

$\frac{3 \left(7 \left(3 x + 2\right) \log{\left(3 x + 2 \right)} - \frac{\sin{\left(14 x \right)}}{2}\right) \tan^{2}{\left(7 x \right)}}{\left(3 x + 2\right) \log{\left(3 x + 2 \right)}^{2} \cos^{2}{\left(7 x \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

   2      /           2     \              3           
tan (7*x)*\21 + 21*tan (7*x)/         3*tan (7*x)      
----------------------------- - -----------------------
         log(3*x + 2)                        2         
                                (3*x + 2)*log (3*x + 2)

$$\frac{\left(21 \tan^{2}{\left(7 x \right)} + 21\right) \tan^{2}{\left(7 x \right)}}{\log{\left(3 x + 2 \right)}} - \frac{3 \tan^{3}{\left(7 x \right)}}{\left(3 x + 2\right) \log{\left(3 x + 2 \right)}^{2}}$$

Simplify

The second derivative [src]

  /                                                                          2      /         2      \\         
  |                                          /       2     \            3*tan (7*x)*|1 + ------------||         
  |   /       2     \ /         2     \   42*\1 + tan (7*x)/*tan(7*x)               \    log(2 + 3*x)/|         
3*|98*\1 + tan (7*x)/*\1 + 2*tan (7*x)/ - --------------------------- + ------------------------------|*tan(7*x)
  |                                          (2 + 3*x)*log(2 + 3*x)                 2                 |         
  \                                                                        (2 + 3*x) *log(2 + 3*x)    /         
----------------------------------------------------------------------------------------------------------------
                                                  log(2 + 3*x)

$$\frac{3 \left(\frac{3 \left(1 + \frac{2}{\log{\left(3 x + 2 \right)}}\right) \tan^{2}{\left(7 x \right)}}{\left(3 x + 2\right)^{2} \log{\left(3 x + 2 \right)}} + 98 \left(\tan^{2}{\left(7 x \right)} + 1\right) \left(2 \tan^{2}{\left(7 x \right)} + 1\right) - \frac{42 \left(\tan^{2}{\left(7 x \right)} + 1\right) \tan{\left(7 x \right)}}{\left(3 x + 2\right) \log{\left(3 x + 2 \right)}}\right) \tan{\left(7 x \right)}}{\log{\left(3 x + 2 \right)}}$$

Simplify

The third derivative [src]

  /                                                                                           3      /         3               3      \                                                                                                    \
  |                                                                                     18*tan (7*x)*|1 + ------------ + -------------|                                                           2      /       2     \ /         2      \|
  |                    /               2                                            \                |    log(2 + 3*x)      2         |       /       2     \ /         2     \            189*tan (7*x)*\1 + tan (7*x)/*|1 + ------------||
  |    /       2     \ |/       2     \         4             2      /       2     \|                \                   log (2 + 3*x)/   882*\1 + tan (7*x)/*\1 + 2*tan (7*x)/*tan(7*x)                                 \    log(2 + 3*x)/|
3*|686*\1 + tan (7*x)/*\\1 + tan (7*x)/  + 2*tan (7*x) + 7*tan (7*x)*\1 + tan (7*x)// - ----------------------------------------------- - ---------------------------------------------- + ------------------------------------------------|
  |                                                                                                          3                                        (2 + 3*x)*log(2 + 3*x)                                    2                          |
  \                                                                                                 (2 + 3*x) *log(2 + 3*x)                                                                            (2 + 3*x) *log(2 + 3*x)             /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                log(2 + 3*x)

$$\frac{3 \left(\frac{189 \left(1 + \frac{2}{\log{\left(3 x + 2 \right)}}\right) \left(\tan^{2}{\left(7 x \right)} + 1\right) \tan^{2}{\left(7 x \right)}}{\left(3 x + 2\right)^{2} \log{\left(3 x + 2 \right)}} + 686 \left(\tan^{2}{\left(7 x \right)} + 1\right) \left(\left(\tan^{2}{\left(7 x \right)} + 1\right)^{2} + 7 \left(\tan^{2}{\left(7 x \right)} + 1\right) \tan^{2}{\left(7 x \right)} + 2 \tan^{4}{\left(7 x \right)}\right) - \frac{882 \left(\tan^{2}{\left(7 x \right)} + 1\right) \left(2 \tan^{2}{\left(7 x \right)} + 1\right) \tan{\left(7 x \right)}}{\left(3 x + 2\right) \log{\left(3 x + 2 \right)}} - \frac{18 \left(1 + \frac{3}{\log{\left(3 x + 2 \right)}} + \frac{3}{\log{\left(3 x + 2 \right)}^{2}}\right) \tan^{3}{\left(7 x \right)}}{\left(3 x + 2\right)^{3} \log{\left(3 x + 2 \right)}}\right)}{\log{\left(3 x + 2 \right)}}$$

Simplify

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Derivative of y=tg^3(7x)/log(3x+2)

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