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Identical expressions
tg(log2(4x^ three - three))
tg( logarithm of 2(4x cubed minus 3))
tg( logarithm of 2(4x to the power of three minus three))
tg(log2(4x3-3))
tglog24x3-3
tg(log2(4x³-3))
tg(log2(4x to the power of 3-3))
tglog24x^3-3
Similar expressions
tg(log2(4x^3+3))

The derivative of the function/ tg(log2(4x^3-3))

Derivative of tg(log2(4x^3-3))

The solution

You have entered [src]

   /   /   3    \\
   |log\4*x  - 3/|
tan|-------------|
   \    log(2)   /

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

tan(log(4*x^3 - 3)/log(2))

Detail solution

Rewrite the function to be differentiated:

$\tan{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)} = \frac{\sin{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\cos{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}$
Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}}$ .
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 $\frac{d}{d x} \frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 4 x^{3} - 3$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x^{3} - 3\right)$ :
      1. Differentiate $4 x^{3} - 3$ term by term:
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{3}$ goes to $3 x^{2}$
        
        So, the result is: $12 x^{2}$
        
        The derivative of the constant $-3$ is zero.
        
        The result is: $12 x^{2}$
      The result of the chain rule is:
      
      $\frac{12 x^{2}}{4 x^{3} - 3}$
    So, the result is: $\frac{12 x^{2}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}$
  The result of the chain rule is:
  
  $\frac{12 x^{2} \cos{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}}$ .
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 $\frac{d}{d x} \frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 4 x^{3} - 3$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x^{3} - 3\right)$ :
      1. Differentiate $4 x^{3} - 3$ term by term:
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{3}$ goes to $3 x^{2}$
        
        So, the result is: $12 x^{2}$
        
        The derivative of the constant $-3$ is zero.
        
        The result is: $12 x^{2}$
      The result of the chain rule is:
      
      $\frac{12 x^{2}}{4 x^{3} - 3}$
    So, the result is: $\frac{12 x^{2}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}$
  The result of the chain rule is:
  
  $- \frac{12 x^{2} \sin{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}$
Now plug in to the quotient rule:

$\frac{\frac{12 x^{2} \sin^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}} + \frac{12 x^{2} \cos^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}}{\cos^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}$
Now simplify:

$\frac{12 x^{2}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)} \cos^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}$

The answer is:

$\frac{12 x^{2}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)} \cos^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                /                         /   /        3\\\
                                |                    3    |log\-3 + 4*x /||
     /        /   /        3\\\ |          3     12*x *tan|--------------||
     |       2|log\-3 + 4*x /|| |       6*x               \    log(2)    /|
24*x*|1 + tan |--------------||*|1 - --------- + -------------------------|
     \        \    log(2)    // |            3       /        3\          |
                                \    -3 + 4*x        \-3 + 4*x /*log(2)   /
---------------------------------------------------------------------------
                             /        3\                                   
                             \-3 + 4*x /*log(2)

$$\frac{24 x \left(\tan^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)} + 1\right) \left(\frac{12 x^{3} \tan{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}} - \frac{6 x^{3}}{4 x^{3} - 3} + 1\right)}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}$$

Simplify

The third derivative [src]

                              /                                         /   /        3\\            /   /        3\\          /        /   /        3\\\              /   /        3\\\
                              |                                    6    |log\-3 + 4*x /|       3    |log\-3 + 4*x /|        6 |       2|log\-3 + 4*x /||        6    2|log\-3 + 4*x /||
   /        /   /        3\\\ |          3             6      432*x *tan|--------------|   72*x *tan|--------------|   144*x *|1 + tan |--------------||   288*x *tan |--------------||
   |       2|log\-3 + 4*x /|| |      36*x         144*x                 \    log(2)    /            \    log(2)    /          \        \    log(2)    //              \    log(2)    /|
24*|1 + tan |--------------||*|1 - --------- + ------------ - -------------------------- + ------------------------- + --------------------------------- + ---------------------------|
   \        \    log(2)    // |            3              2                 2                  /        3\                               2                                2           |
                              |    -3 + 4*x    /        3\       /        3\                   \-3 + 4*x /*log(2)             /        3\     2                /        3\     2      |
                              \                \-3 + 4*x /       \-3 + 4*x / *log(2)                                          \-3 + 4*x / *log (2)             \-3 + 4*x / *log (2)   /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                   /        3\                                                                                         
                                                                                   \-3 + 4*x /*log(2)

$$\frac{24 \left(\tan^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)} + 1\right) \left(\frac{144 x^{6} \left(\tan^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)} + 1\right)}{\left(4 x^{3} - 3\right)^{2} \log{\left(2 \right)}^{2}} + \frac{288 x^{6} \tan^{2}{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right)^{2} \log{\left(2 \right)}^{2}} - \frac{432 x^{6} \tan{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right)^{2} \log{\left(2 \right)}} + \frac{144 x^{6}}{\left(4 x^{3} - 3\right)^{2}} + \frac{72 x^{3} \tan{\left(\frac{\log{\left(4 x^{3} - 3 \right)}}{\log{\left(2 \right)}} \right)}}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}} - \frac{36 x^{3}}{4 x^{3} - 3} + 1\right)}{\left(4 x^{3} - 3\right) \log{\left(2 \right)}}$$

Simplify

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Derivative of tg(log2(4x^3-3))

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