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Identical expressions
(ln(5x- three))/(four tg3x^4)
(ln(5x minus 3)) divide by (4tg3x to the power of 4)
(ln(5x minus three)) divide by (four tg3x to the power of 4)
(ln(5x-3))/(4tg3x4)
ln5x-3/4tg3x4
(ln(5x-3))/(4tg3x⁴)
ln5x-3/4tg3x^4
(ln(5x-3)) divide by (4tg3x^4)
Similar expressions
y=ln(5x-3)/(4tg(3x^4))
(ln(5x+3))/(4tg3x^4)

The derivative of the function/ (ln(5x-3))/(4tg3x^4)

Derivative of (ln(5x-3))/(4tg3x^4)

The solution

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log(5*x - 3)
------------
     4      
4*tan (3*x)

$$\frac{\log{\left(5 x - 3 \right)}}{4 \tan^{4}{\left(3 x \right)}}$$

log(5*x - 3)/((4*tan(3*x)^4))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(5 x - 3 \right)}$ and $g{\left(x \right)} = 4 \tan^{4}{\left(3 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 5 x - 3$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x - 3\right)$ :
  1. Differentiate $5 x - 3$ 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: $5$
    2. The derivative of the constant $-3$ is zero.
    The result is: $5$
  The result of the chain rule is:
  
  $\frac{5}{5 x - 3}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \tan{\left(3 x \right)}$ .
  2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 x \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Let $u = 3 x$ .
      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 $\frac{d}{d x} 3 x$ :
        
        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: $3$
        
        The result of the chain rule is:
        
        $3 \cos{\left(3 x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Let $u = 3 x$ .
      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 $\frac{d}{d x} 3 x$ :
        
        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: $3$
        
        The result of the chain rule is:
        
        $- 3 \sin{\left(3 x \right)}$
      Now plug in to the quotient rule:
      
      $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
    The result of the chain rule is:
    
    $\frac{4 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \tan^{3}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
  So, the result is: $\frac{16 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \tan^{3}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}}$
Now plug in to the quotient rule:

$\frac{- \frac{16 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) \log{\left(5 x - 3 \right)} \tan^{3}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{20 \tan^{4}{\left(3 x \right)}}{5 x - 3}}{16 \tan^{8}{\left(3 x \right)}}$
Now simplify:

$\frac{- 3 \left(5 x - 3\right) \log{\left(5 x - 3 \right)} + \frac{5 \sin{\left(6 x \right)}}{8}}{\left(5 x - 3\right) \cos^{2}{\left(3 x \right)} \tan^{5}{\left(3 x \right)}}$

The answer is:

$\frac{- 3 \left(5 x - 3\right) \log{\left(5 x - 3 \right)} + \frac{5 \sin{\left(6 x \right)}}{8}}{\left(5 x - 3\right) \cos^{2}{\left(3 x \right)} \tan^{5}{\left(3 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       1                                        
5*-----------                                   
       4        /           2     \             
  4*tan (3*x)   \12 + 12*tan (3*x)/*log(5*x - 3)
------------- - --------------------------------
   5*x - 3                     5                
                          4*tan (3*x)

$$- \frac{\left(12 \tan^{2}{\left(3 x \right)} + 12\right) \log{\left(5 x - 3 \right)}}{4 \tan^{5}{\left(3 x \right)}} + \frac{5 \frac{1}{4 \tan^{4}{\left(3 x \right)}}}{5 x - 3}$$

Simplify

The second derivative [src]

                      /       2     \                     /       /       2     \\              
        25         30*\1 + tan (3*x)/     /       2     \ |     5*\1 + tan (3*x)/|              
- ------------- - ------------------- + 9*\1 + tan (3*x)/*|-2 + -----------------|*log(-3 + 5*x)
              2   (-3 + 5*x)*tan(3*x)                     |            2         |              
  4*(-3 + 5*x)                                            \         tan (3*x)    /              
------------------------------------------------------------------------------------------------
                                              4                                                 
                                           tan (3*x)

$$\frac{9 \left(\frac{5 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan^{2}{\left(3 x \right)}} - 2\right) \left(\tan^{2}{\left(3 x \right)} + 1\right) \log{\left(5 x - 3 \right)} - \frac{30 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\left(5 x - 3\right) \tan{\left(3 x \right)}} - \frac{25}{4 \left(5 x - 3\right)^{2}}}{\tan^{4}{\left(3 x \right)}}$$

Simplify

The third derivative [src]

                                                                                                                                                       /       /       2     \\
                                                                                                                                       /       2     \ |     5*\1 + tan (3*x)/|
                                            /                                           2\                                         135*\1 + tan (3*x)/*|-2 + -----------------|
                                            |       /       2     \      /       2     \ |                      /       2     \                        |            2         |
         125                /       2     \ |    14*\1 + tan (3*x)/   15*\1 + tan (3*x)/ |                  225*\1 + tan (3*x)/                        \         tan (3*x)    /
---------------------- - 54*\1 + tan (3*x)/*|2 - ------------------ + -------------------|*log(-3 + 5*x) + --------------------- + --------------------------------------------
            3                               |           2                     4          |                           2    2                    (-3 + 5*x)*tan(3*x)             
2*(-3 + 5*x) *tan(3*x)                      \        tan (3*x)             tan (3*x)     /                 (-3 + 5*x) *tan (3*x)                                               
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                      3                                                                                        
                                                                                   tan (3*x)

$$\frac{- 54 \left(\tan^{2}{\left(3 x \right)} + 1\right) \left(\frac{15 \left(\tan^{2}{\left(3 x \right)} + 1\right)^{2}}{\tan^{4}{\left(3 x \right)}} - \frac{14 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan^{2}{\left(3 x \right)}} + 2\right) \log{\left(5 x - 3 \right)} + \frac{135 \left(\frac{5 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\tan^{2}{\left(3 x \right)}} - 2\right) \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\left(5 x - 3\right) \tan{\left(3 x \right)}} + \frac{225 \left(\tan^{2}{\left(3 x \right)} + 1\right)}{\left(5 x - 3\right)^{2} \tan^{2}{\left(3 x \right)}} + \frac{125}{2 \left(5 x - 3\right)^{3} \tan{\left(3 x \right)}}}{\tan^{3}{\left(3 x \right)}}$$

Simplify

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Derivative of (ln(5x-3))/(4tg3x^4)

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