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Similar expressions
y=x(coslnx-sinlnx)/2
x*(cos(ln(x))+sin(ln(x)))/2

The derivative of the function/ y=x(coslnx+sinlnx)/2

Derivative of y=x(coslnx+sinlnx)/2

The solution

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)}$ term by term:
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      $g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}$
    2. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      $g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
    The result is: $- \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x}$
  The result is: $x \left(- \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x}\right) + \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)}$
So, the result is: $\frac{x \left(- \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x}\right)}{2} + \frac{\log{\left(x \right)} \sin{\left(x \right)}}{2} + \frac{\log{\left(x \right)} \cos{\left(x \right)}}{2}$
Now simplify:

$\frac{\sqrt{2} \left(x \log{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)} + \log{\left(x \right)} \sin{\left(x + \frac{\pi}{4} \right)} + \sin{\left(x + \frac{\pi}{4} \right)}\right)}{2}$

The answer is:

$\frac{\sqrt{2} \left(x \log{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)} + \log{\left(x \right)} \sin{\left(x + \frac{\pi}{4} \right)} + \sin{\left(x + \frac{\pi}{4} \right)}\right)}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=x(coslnx+sinlnx)/2

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The solution