Derivative of ln(tgx/2)-x/sinx

The solution

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

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

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

Detail solution

Differentiate $- \frac{x}{\sin{\left(x \right)}} + \log{\left(\frac{\tan{\left(x \right)}}{2} \right)}$ term by term:
1. Let $u = \frac{\tan{\left(x \right)}}{2}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\tan{\left(x \right)}}{2}$ :
  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 $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. 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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
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: $- \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(tgx/2)-x/sinx

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