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Identical expressions
tg*(ln*(tg(sqrt(three)*x- one)))
tg multiply by (ln multiply by (tg( square root of (3) multiply by x minus 1)))
tg multiply by (ln multiply by (tg( square root of (three) multiply by x minus one)))
tg*(ln*(tg(√(3)*x-1)))
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Similar expressions
tg*ln*(tg(sqrt(3)*x-1))
tg*(ln*(tg(sqrt(3)*x+1)))

The derivative of the function/ tg*(ln*(tg(sqrt(3)*x-1)))

Derivative of tg(ln(tg(sqrt(3)*x-1)))

The solution

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   /   /   /  ___      \\\
tan\log\tan\\/ 3 *x - 1///

$$\tan{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}$$

tan(log(tan(sqrt(3)*x - 1)))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} = \frac{\sin{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}$ .

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

$\frac{\frac{\left(\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}\right) \sin^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}} + \frac{\left(\sqrt{3} \sin^{2}{\left(\sqrt{3} x - 1 \right)} + \sqrt{3} \cos^{2}{\left(\sqrt{3} x - 1 \right)}\right) \cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \tan{\left(\sqrt{3} x - 1 \right)}}}{\cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)}}$
Now simplify:

$\frac{\sqrt{3}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} \tan{\left(\sqrt{3} x - 1 \right)}}$

The answer is:

$\frac{\sqrt{3}}{\cos^{2}{\left(\sqrt{3} x - 1 \right)} \cos^{2}{\left(\log{\left(\tan{\left(\sqrt{3} x - 1 \right)} \right)} \right)} \tan{\left(\sqrt{3} x - 1 \right)}}$

The graph

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Plot the graph f'(x)

The first derivative [src]

  ___ /       2/  ___      \\ /       2/   /   /  ___      \\\\
\/ 3 *\1 + tan \\/ 3 *x - 1//*\1 + tan \log\tan\\/ 3 *x - 1////
---------------------------------------------------------------
                           /  ___      \                       
                        tan\\/ 3 *x - 1/

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

Simplify

The second derivative [src]

                                                              /           2/         ___\     /       2/         ___\\    /   /   /         ___\\\\
  /       2/         ___\\ /       2/   /   /         ___\\\\ |    1 + tan \-1 + x*\/ 3 /   2*\1 + tan \-1 + x*\/ 3 //*tan\log\tan\-1 + x*\/ 3 ///|
3*\1 + tan \-1 + x*\/ 3 //*\1 + tan \log\tan\-1 + x*\/ 3 ////*|2 - ---------------------- + ------------------------------------------------------|
                                                              |         2/         ___\                          2/         ___\                  |
                                                              \      tan \-1 + x*\/ 3 /                       tan \-1 + x*\/ 3 /                  /

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

Simplify

The third derivative [src]

                                                                    /                                              2                                                        2                                                                2                                                         2                                                                                      \
                                                                    |                      /       2/         ___\\      /       2/         ___\\   /       2/         ___\\  /       2/   /   /         ___\\\\     /       2/         ___\\     /   /   /         ___\\\     /       2/         ___\\     2/   /   /         ___\\\     /       2/         ___\\    /   /   /         ___\\\|
    ___ /       2/         ___\\ /       2/   /   /         ___\\\\ |     /         ___\   \1 + tan \-1 + x*\/ 3 //    2*\1 + tan \-1 + x*\/ 3 //   \1 + tan \-1 + x*\/ 3 // *\1 + tan \log\tan\-1 + x*\/ 3 ////   3*\1 + tan \-1 + x*\/ 3 // *tan\log\tan\-1 + x*\/ 3 ///   2*\1 + tan \-1 + x*\/ 3 // *tan \log\tan\-1 + x*\/ 3 ///   6*\1 + tan \-1 + x*\/ 3 //*tan\log\tan\-1 + x*\/ 3 ///|
6*\/ 3 *\1 + tan \-1 + x*\/ 3 //*\1 + tan \log\tan\-1 + x*\/ 3 ////*|2*tan\-1 + x*\/ 3 / + ------------------------- - -------------------------- + ------------------------------------------------------------ - ------------------------------------------------------- + -------------------------------------------------------- + ------------------------------------------------------|
                                                                    |                             3/         ___\             /         ___\                                3/         ___\                                              3/         ___\                                           3/         ___\                                           /         ___\                   |
                                                                    \                          tan \-1 + x*\/ 3 /          tan\-1 + x*\/ 3 /                             tan \-1 + x*\/ 3 /                                           tan \-1 + x*\/ 3 /                                        tan \-1 + x*\/ 3 /                                        tan\-1 + x*\/ 3 /                   /

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

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Derivative of tg(ln(tg(sqrt(3)*x-1)))

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Derivative of tg*(ln*(tg(sqrt(3)*x-1)))

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Derivative of tg(ln(tg(sqrt(3)*x-1)))