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Derivative of log(log(x))
Identical expressions
y=sin(logx)-cos(logx)/x
y equally sinus of ( logarithm of x) minus co sinus of e of ( logarithm of x) divide by x
y=sinlogx-coslogx/x
y=sin(logx)-cos(logx) divide by x
Similar expressions
y=sin(logx)+cos(logx)/x

The derivative of the function/ y=sin(logx)-cos(logx)/x

Derivative of y=sin(logx)-cos(logx)/x

The solution

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              cos(log(x))
sin(log(x)) - -----------
                   x

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

d /              cos(log(x))\
--|sin(log(x)) - -----------|
dx\                   x     /

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

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Detail solution

Differentiate $\sin{\left(\log{\left(x \right)} \right)} - \frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$ term by term:
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:
  
  $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
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:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(\log{\left(x \right)} \right)}$ and $g{\left(x \right)} = x$ .
    
    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:
      
      $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    Now plug in to the quotient rule:
    
    $\frac{- \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}$
  So, the result is: $- \frac{- \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}$
The result is: $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x} - \frac{- \sin{\left(\log{\left(x \right)} \right)} - \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

$$\frac{3 \sin{\left(\log{\left(x \right)} \right)} + \cos{\left(\log{\left(x \right)} \right)} + \frac{10 \sin{\left(\log{\left(x \right)} \right)}}{x}}{x^{3}}$$

Simplify

The graph

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Derivative of y=sin(logx)-cos(logx)/x

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