Derivative of log4log2tgx

The solution

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log(4*log(2*tan(x)))

$$\log{\left(4 \log{\left(2 \tan{\left(x \right)} \right)} \right)}$$

log(4*log(2*tan(x)))

Detail solution

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

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

$\frac{2}{\log{\left(2 \tan{\left(x \right)} \right)} \sin{\left(2 x \right)}}$

The answer is:

$\frac{2}{\log{\left(2 \tan{\left(x \right)} \right)} \sin{\left(2 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

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Derivative of log4log2tgx

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