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Derivative of:
Derivative of 2^(3*x)
Derivative of -2/x
Derivative of sin(x)*cos(x)
Derivative of 1/cos(x)
Identical expressions
y=ln*tg(x/ two)-(x/sinx)
y equally ln multiply by tg(x divide by 2) minus (x divide by sinus of x)
y equally ln multiply by tg(x divide by two) minus (x divide by sinus of x)
y=lntg(x/2)-(x/sinx)
y=lntgx/2-x/sinx
y=ln*tg(x divide by 2)-(x divide by sinx)
Similar expressions
y=ln*tg(x/2)+(x/sinx)
y=(lntg(x/2))-(x/sinx)
ln(tgx/2)-x/sinx
y=ln(tg(x/2))-x/sinx
y=lntg(x/2)-x/sinx

The derivative of the function/ y=ln*tg(x/2)-(x/sinx)

Derivative of y=ln*tg(x/2)-(x/sinx)

The solution

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

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

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

Detail solution

Differentiate $- \frac{x}{\sin{\left(x \right)}} + \log{\left(x \right)} \tan{\left(\frac{x}{2} \right)}$ term by term:
1. 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 $\frac{1}{x}$ .
  $g{\left(x \right)} = \tan{\left(\frac{x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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 is: $\frac{\left(\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}\right) \log{\left(x \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{x}$
2. 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: $- \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\left(\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}\right) \log{\left(x \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=ln*tg(x/2)-(x/sinx)

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