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(-ln2)cos(logx)²
The derivative of the function/ (-ln2)cos(logx)²

Derivative of (-ln2)cos(logx)²

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

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The solution

You have entered [src] 
           2        
-log(2)*cos (log(x))
−log⁡(2)cos⁡2(log⁡(x))- \log{\left(2 \right)} \cos^{2}{\left(\log{\left(x \right)} \right)} 
d /           2        \
--\-log(2)*cos (log(x))/
dx
ddx−log⁡(2)cos⁡2(log⁡(x))\frac{d}{d x} - \log{\left(2 \right)} \cos^{2}{\left(\log{\left(x \right)} \right)}
Transform
Detail solution

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

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

    2. Apply the power rule: u2u^{2} goes to 2u2 u

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

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

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

        The result of the chain rule is:

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

      The result of the chain rule is:

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

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

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
2*cos(log(x))*log(2)*sin(log(x))
--------------------------------
               x
2log⁡(2)sin⁡(log⁡(x))cos⁡(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
−2(sin⁡2(log⁡(x))+sin⁡(log⁡(x))cos⁡(log⁡(x))−cos⁡2(log⁡(x)))log⁡(2)x2- \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
−2(−3sin⁡2(log⁡(x))+2sin⁡(log⁡(x))cos⁡(log⁡(x))+3cos⁡2(log⁡(x)))log⁡(2)x3- \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
Derivative of (-ln2)cos(logx)²
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