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y=lntg(x/ two)-x/sinx
y equally lntg(x divide by 2) minus x divide by sinus of x
y equally lntg(x divide by two) minus x divide by sinus of x
y=lntgx/2-x/sinx
y=lntg(x divide by 2)-x divide by sinx
Similar expressions
y=ln(tg(x/2))-(x/sin(x))
y=(lntg(x/2))-(x/sinx)
y=lntg(x/2)+x/sinx
y=ln(tg(x/2))-x/sinx

The derivative of the function/ y=lntg(x/2)-x/sinx

Derivative of y=lntg(x/2)-x/sinx

The solution

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   /   /x\\     x   
log|tan|-|| - ------
   \   \2//   sin(x)

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

log(tan(x/2)) - x/sin(x)

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                           2                3                           
           /       2/x\\    /x\               /       2/x\\    /       2/x\\                            
           |1 + tan |-||*tan|-|        2      |1 + tan |-||    |1 + tan |-||                        3   
    3      \        \2//    \2/   6*cos (x)   \        \2//    \        \2//    5*x*cos(x)   6*x*cos (x)
- ------ + -------------------- - --------- - -------------- + -------------- + ---------- + -----------
  sin(x)            2                 3               /x\             3/x\          2             4     
                                   sin (x)       2*tan|-|        4*tan |-|       sin (x)       sin (x)  
                                                      \2/              \2/

$$\frac{5 x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{6 x \cos^{3}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{4 \tan^{3}{\left(\frac{x}{2} \right)}} - \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{2 \tan{\left(\frac{x}{2} \right)}} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2} - \frac{3}{\sin{\left(x \right)}} - \frac{6 \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}}$$

Simplify

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Derivative of y=lntg(x/2)-x/sinx

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