Mister Exam

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)/
$$\cos{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}$$
cos(log(x)/log(2))
Detail solution
  1. Let .

  2. The derivative of cosine is negative sine:

  3. Then, apply the chain rule. Multiply by :

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

      1. The derivative of is .

      So, the result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
    /log(x)\ 
-sin|------| 
    \log(2)/ 
-------------
   x*log(2)  
$$- \frac{\sin{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}}{x \log{\left(2 \right)}}$$
The second derivative [src]
     /log(x)\              
  cos|------|              
     \log(2)/      /log(x)\
- ----------- + sin|------|
     log(2)        \log(2)/
---------------------------
          2                
         x *log(2)         
$$\frac{\sin{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)} - \frac{\cos{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}}{\log{\left(2 \right)}}}{x^{2} \log{\left(2 \right)}}$$
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)                  
$$\frac{- 2 \sin{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)} + \frac{\sin{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}}{\log{\left(2 \right)}^{2}} + \frac{3 \cos{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}}{\log{\left(2 \right)}}}{x^{3} \log{\left(2 \right)}}$$