Derivative of cos(logx/log2)

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   /log(x)\
cos|------|
   \log(2)/

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

cos(log(x)/log(2))

Detail solution

Let $u = \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  So, the result is: $\frac{1}{x \log{\left(2 \right)}}$
The result of the chain rule is:

$- \frac{\sin{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}}{x \log{\left(2 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

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)

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

Simplify

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Derivative of cos(logx/log2)

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The solution