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cosx/lnx
The derivative of the function/ cosx/lnx

Derivative of cosx/lnx

Function f() - derivative -N order at the point
v

The graph:

from to

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The solution

You have entered [src] 
cos(x)
------
log(x)
cos⁡(x)log⁡(x)\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} 
cos(x)/log(x)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} and g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    Now plug in to the quotient rule:

    −log⁡(x)sin⁡(x)−cos⁡(x)xlog⁡(x)2\frac{- \log{\left(x \right)} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}}

  2. Now simplify:

    −sin⁡(x)log⁡(x)−cos⁡(x)xlog⁡(x)2- \frac{\sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x \log{\left(x \right)}^{2}}

The answer is:

−sin⁡(x)log⁡(x)−cos⁡(x)xlog⁡(x)2- \frac{\sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x \log{\left(x \right)}^{2}}

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  sin(x)     cos(x) 
- ------ - ---------
  log(x)        2   
           x*log (x)
−sin⁡(x)log⁡(x)−cos⁡(x)xlog⁡(x)2- \frac{\sin{\left(x \right)}}{\log{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x \log{\left(x \right)}^{2}}
Simplify
The second derivative [src] 
                     /      2   \       
                     |1 + ------|*cos(x)
          2*sin(x)   \    log(x)/       
-cos(x) + -------- + -------------------
          x*log(x)         2            
                          x *log(x)     
----------------------------------------
                 log(x)
−cos⁡(x)+2sin⁡(x)xlog⁡(x)+(1+2log⁡(x))cos⁡(x)x2log⁡(x)log⁡(x)\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}
Simplify
The third derivative [src] 
                                     /      3         3   \                
             /      2   \          2*|1 + ------ + -------|*cos(x)         
           3*|1 + ------|*sin(x)     |    log(x)      2   |                
3*cos(x)     \    log(x)/            \             log (x)/                
-------- - --------------------- - ------------------------------- + sin(x)
x*log(x)          2                            3                           
                 x *log(x)                    x *log(x)                    
---------------------------------------------------------------------------
                                   log(x)
sin⁡(x)+3cos⁡(x)xlog⁡(x)−3(1+2log⁡(x))sin⁡(x)x2log⁡(x)−2(1+3log⁡(x)+3log⁡(x)2)cos⁡(x)x3log⁡(x)log⁡(x)\frac{\sin{\left(x \right)} + \frac{3 \cos{\left(x \right)}}{x \log{\left(x \right)}} - \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \cos{\left(x \right)}}{x^{3} \log{\left(x \right)}}}{\log{\left(x \right)}}
Simplify
The graph
Derivative of cosx/lnx
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