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Similar expressions
сos(x)^2-log(tan(x/2))

The derivative of the function/ сos(x)^2+log(tan(x/2))

Derivative of сos(x)^2+log(tan(x/2))

The solution

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

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

cos(x)^2 + log(tan(x/2))

Detail solution

Differentiate $\log{\left(\tan{\left(\frac{x}{2} \right)} \right)} + \cos^{2}{\left(x \right)}$ term by term:
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
4. Let $u = \tan{\left(\frac{x}{2} \right)}$ .
5. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
6. 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:
  
  $\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}}$
The result is: $\frac{\frac{\sin^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{\cos^{2}{\left(\frac{x}{2} \right)}}{2}}{\cos^{2}{\left(\frac{x}{2} \right)} \tan{\left(\frac{x}{2} \right)}} - 2 \sin{\left(x \right)} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                      2                3
/       2/x\\    /x\                     /       2/x\\    /       2/x\\ 
|1 + tan |-||*tan|-|                     |1 + tan |-||    |1 + tan |-|| 
\        \2//    \2/                     \        \2//    \        \2// 
-------------------- + 8*cos(x)*sin(x) - -------------- + --------------
         2                                       /x\             3/x\   
                                            2*tan|-|        4*tan |-|   
                                                 \2/              \2/

$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}{4 \tan^{3}{\left(\frac{x}{2} \right)}} - \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{2 \tan{\left(\frac{x}{2} \right)}} + \frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{2} + 8 \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplify

The graph

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Derivative of сos(x)^2+log(tan(x/2))

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