Derivative of cos1/log2*x

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 cos(1) 
--------
log(2*x)

$$\frac{\cos{\left(1 \right)}}{\log{\left(2 x \right)}}$$

cos(1)/log(2*x)

Detail solution

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

The answer is:

$- \frac{\cos{\left(1 \right)}}{x \log{\left(2 x \right)}^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -cos(1)  
-----------
     2     
x*log (2*x)

$$- \frac{\cos{\left(1 \right)}}{x \log{\left(2 x \right)}^{2}}$$

Simplify

The second derivative [src]

/       2    \       
|1 + --------|*cos(1)
\    log(2*x)/       
---------------------
      2    2         
     x *log (2*x)

$$\frac{\left(1 + \frac{2}{\log{\left(2 x \right)}}\right) \cos{\left(1 \right)}}{x^{2} \log{\left(2 x \right)}^{2}}$$

Simplify

The third derivative [src]

   /       3           3    \       
-2*|1 + -------- + ---------|*cos(1)
   |    log(2*x)      2     |       
   \               log (2*x)/       
------------------------------------
             3    2                 
            x *log (2*x)

$$- \frac{2 \left(1 + \frac{3}{\log{\left(2 x \right)}} + \frac{3}{\log{\left(2 x \right)}^{2}}\right) \cos{\left(1 \right)}}{x^{3} \log{\left(2 x \right)}^{2}}$$

Simplify

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Derivative of cos1/log2*x

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The solution