Derivative of (-ln2)cos(logx)²

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

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

d /           2        \
--\-log(2)*cos (log(x))/
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   /   2              2                                  \       
-2*\sin (log(x)) - cos (log(x)) + cos(log(x))*sin(log(x))/*log(2)
-----------------------------------------------------------------
                                 2                               
                                x

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (-ln2)cos(logx)²

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The solution