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Identical expressions
y=ln*tg((2x+ one)/ four)
y equally ln multiply by tg((2x plus 1) divide by 4)
y equally ln multiply by tg((2x plus one) divide by four)
y=lntg((2x+1)/4)
y=lntg2x+1/4
y=ln*tg((2x+1) divide by 4)
Similar expressions
y=ln*tg((2x-1)/4)
y=ln(tg2x+1/4)
ln*tg(2x+1/4)

The derivative of the function/ y=ln*tg((2x+1)/4)

Derivative of y=ln*tg((2x+1)/4)

The solution

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

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

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

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \tan{\left(\frac{2 x + 1}{4} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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.
      1. 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.
      1. 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 is: $\frac{\left(\frac{\sin^{2}{\left(\frac{2 x + 1}{4} \right)}}{2} + \frac{\cos^{2}{\left(\frac{2 x + 1}{4} \right)}}{2}\right) \log{\left(x \right)}}{\cos^{2}{\left(\frac{2 x + 1}{4} \right)}} + \frac{\tan{\left(\frac{2 x + 1}{4} \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=ln*tg((2x+1)/4)

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