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Identical expressions
сos(ln(x))((ln(x))/ two)+(((ln(x))^ two)/ two)sin(ln(x))
сos(ln(x))((ln(x)) divide by 2) plus (((ln(x)) squared ) divide by 2) sinus of (ln(x))
сos(ln(x))((ln(x)) divide by two) plus (((ln(x)) to the power of two) divide by two) sinus of (ln(x))
сos(ln(x))((ln(x))/2)+(((ln(x))2)/2)sin(ln(x))
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сoslnxlnx/2+lnx^2/2sinlnx
сos(ln(x))((ln(x)) divide by 2)+(((ln(x))^2) divide by 2)sin(ln(x))
Similar expressions
сos(ln(x))((ln(x))/2)-(((ln(x))^2)/2)sin(ln(x))

The derivative of the function/ сos(ln(x))((ln(x))/2)+(((ln(x))^2)/2)sin(ln(x))

Derivative of сos(ln(x))((ln(x))/2)+(((ln(x))^2)/2)sin(ln(x))

The solution

You have entered [src]

                        2               
            log(x)   log (x)            
cos(log(x))*------ + -------*sin(log(x))
              2         2

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

cos(log(x))*(log(x)/2) + (log(x)^2/2)*sin(log(x))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 2                                    
cos(log(x))   log (x)*cos(log(x))   log(x)*sin(log(x))
----------- + ------------------- + ------------------
    2*x               2*x                  2*x

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

Simplify

The second derivative [src]

                  2                     2                                                           
-cos(log(x)) - log (x)*cos(log(x)) - log (x)*sin(log(x)) - log(x)*sin(log(x)) + 3*cos(log(x))*log(x)
----------------------------------------------------------------------------------------------------
                                                   2                                                
                                                2*x

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

Simplify

The third derivative [src]

                   2                                                                     2               
5*cos(log(x)) + log (x)*cos(log(x)) - 9*cos(log(x))*log(x) - 3*log(x)*sin(log(x)) + 3*log (x)*sin(log(x))
---------------------------------------------------------------------------------------------------------
                                                      3                                                  
                                                   2*x

$$\frac{3 \log{\left(x \right)}^{2} \sin{\left(\log{\left(x \right)} \right)} + \log{\left(x \right)}^{2} \cos{\left(\log{\left(x \right)} \right)} - 3 \log{\left(x \right)} \sin{\left(\log{\left(x \right)} \right)} - 9 \log{\left(x \right)} \cos{\left(\log{\left(x \right)} \right)} + 5 \cos{\left(\log{\left(x \right)} \right)}}{2 x^{3}}$$

Simplify

The graph

Derivative of сos(ln(x))((ln(x))/2)+(((ln(x))^2)/2)sin(ln(x))

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How to use it?

Identical expressions

Similar expressions

Derivative of сos(ln(x))((ln(x))/2)+(((ln(x))^2)/2)sin(ln(x))

The graph:

Piecewise:

The solution