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The derivative of the function/ cos(logx/log2)

Derivative of cos(logx/log2)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /log(x)\
cos|------|
   \log(2)/
cos⁡(log⁡(x)log⁡(2))\cos{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)} 
cos(log(x)/log(2))
Detail solution

  1. Let u=log⁡(x)log⁡(2)u = \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}.

  2. The derivative of cosine is negative sine:

    dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

  3. Then, apply the chain rule. Multiply by ddxlog⁡(x)log⁡(2)\frac{d}{d x} \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}:

    1. The derivative of a constant times a function is the constant times the derivative of the function.

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

      So, the result is: 1xlog⁡(2)\frac{1}{x \log{\left(2 \right)}}

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

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

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