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Similar expressions
log(tan((2*x-1)/4))^(2)

The derivative of the function/ log(tan((2*x+1)/4))^(2)

Derivative of log(tan((2*x+1)/4))^(2)

The solution

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   2/   /2*x + 1\\
log |tan|-------||
    \   \   4   //

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

log(tan((2*x + 1)/4))^2

Detail solution

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

$\frac{2 \left(\frac{\sin^{2}{\left(\frac{2 x + 1}{4} \right)}}{2} + \frac{\cos^{2}{\left(\frac{2 x + 1}{4} \right)}}{2}\right) \log{\left(\tan{\left(\frac{2 x + 1}{4} \right)} \right)}}{\cos^{2}{\left(\frac{2 x + 1}{4} \right)} \tan{\left(\frac{2 x + 1}{4} \right)}}$
Now simplify:

$\frac{2 \log{\left(\tan{\left(\frac{x}{2} + \frac{1}{4} \right)} \right)}}{\left(\cos{\left(x + \frac{1}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{1}{4} \right)}}$

The answer is:

$\frac{2 \log{\left(\tan{\left(\frac{x}{2} + \frac{1}{4} \right)} \right)}}{\left(\cos{\left(x + \frac{1}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} + \frac{1}{4} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /       2/2*x + 1\\                  
  |    tan |-------||                  
  |1       \   4   /|    /   /2*x + 1\\
2*|- + -------------|*log|tan|-------||
  \2         2      /    \   \   4   //
---------------------------------------
                 /2*x + 1\             
              tan|-------|             
                 \   4   /

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

Simplify

The second derivative [src]

/       2/1 + 2*x\\ /                             2/1 + 2*x\   /       2/1 + 2*x\\    /   /1 + 2*x\\\
|    tan |-------|| |                      1 + tan |-------|   |1 + tan |-------||*log|tan|-------|||
|1       \   4   /| |     /   /1 + 2*x\\           \   4   /   \        \   4   //    \   \   4   //|
|- + -------------|*|2*log|tan|-------|| + ----------------- - -------------------------------------|
\2         2      / |     \   \   4   //        2/1 + 2*x\                    2/1 + 2*x\            |
                    |                        tan |-------|                 tan |-------|            |
                    \                            \   4   /                     \   4   /            /

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(tan((2*x+1)/4))^(2)

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