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The derivative of the function/ (log2(sinx^2))^3

Derivative of (log2(sinx^2))^3

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

The graph:

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The solution

You have entered [src] 
              3
/   /   2   \\ 
|log\sin (x)/| 
|------------| 
\   log(2)   /
(log⁡(sin⁡2(x))log⁡(2))3\left(\frac{\log{\left(\sin^{2}{\left(x \right)} \right)}}{\log{\left(2 \right)}}\right)^{3} 
(log(sin(x)^2)/log(2))^3
Detail solution

  1. Let u=log⁡(sin⁡2(x))log⁡(2)u = \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}}{\log{\left(2 \right)}}.

  2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

  3. Then, apply the chain rule. Multiply by ddxlog⁡(sin⁡2(x))log⁡(2)\frac{d}{d x} \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}}{\log{\left(2 \right)}}:

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

      1. Let u=sin⁡2(x)u = \sin^{2}{\left(x \right)}.

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

      3. Then, apply the chain rule. Multiply by ddxsin⁡2(x)\frac{d}{d x} \sin^{2}{\left(x \right)}:

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

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

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

          1. The derivative of sine is cosine:

            ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

          The result of the chain rule is:

          2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

        The result of the chain rule is:

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

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

    The result of the chain rule is:

    6log⁡(sin⁡2(x))2cos⁡(x)log⁡(2)3sin⁡(x)\frac{6 \log{\left(\sin^{2}{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{\log{\left(2 \right)}^{3} \sin{\left(x \right)}}

  4. Now simplify:

    6log⁡(sin⁡2(x))2log⁡(2)3tan⁡(x)\frac{6 \log{\left(\sin^{2}{\left(x \right)} \right)}^{2}}{\log{\left(2 \right)}^{3} \tan{\left(x \right)}}

The answer is:

6log⁡(sin⁡2(x))2log⁡(2)3tan⁡(x)\frac{6 \log{\left(\sin^{2}{\left(x \right)} \right)}^{2}}{\log{\left(2 \right)}^{3} \tan{\left(x \right)}}

The graph
02468-8-6-4-2-1010-500000500000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
     3/   2   \       
  log \sin (x)/       
6*-------------*cos(x)
        3             
     log (2)          
----------------------
    /   2   \         
 log\sin (x)/*sin(x)
6log⁡(sin⁡2(x))3log⁡(2)3cos⁡(x)log⁡(sin⁡2(x))sin⁡(x)\frac{6 \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}^{3}}{\log{\left(2 \right)}^{3}} \cos{\left(x \right)}}{\log{\left(\sin^{2}{\left(x \right)} \right)} \sin{\left(x \right)}}
Simplify
The second derivative [src] 
  /                      2         2       /   2   \\             
  |     /   2   \   4*cos (x)   cos (x)*log\sin (x)/|    /   2   \
6*|- log\sin (x)/ + --------- - --------------------|*log\sin (x)/
  |                     2                2          |             
  \                  sin (x)          sin (x)       /             
------------------------------------------------------------------
                                3                                 
                             log (2)
6(−log⁡(sin⁡2(x))−log⁡(sin⁡2(x))cos⁡2(x)sin⁡2(x)+4cos⁡2(x)sin⁡2(x))log⁡(sin⁡2(x))log⁡(2)3\frac{6 \left(- \log{\left(\sin^{2}{\left(x \right)} \right)} - \frac{\log{\left(\sin^{2}{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\sin^{2}{\left(x \right)} \right)}}{\log{\left(2 \right)}^{3}}
Simplify
The third derivative [src] 
   /                                      2         2       2/   2   \        2       /   2   \\       
   |   2/   2   \        /   2   \   4*cos (x)   cos (x)*log \sin (x)/   6*cos (x)*log\sin (x)/|       
12*|log \sin (x)/ - 6*log\sin (x)/ + --------- + --------------------- - ----------------------|*cos(x)
   |                                     2                 2                       2           |       
   \                                  sin (x)           sin (x)                 sin (x)        /       
-------------------------------------------------------------------------------------------------------
                                                3                                                      
                                             log (2)*sin(x)
12(log⁡(sin⁡2(x))2+log⁡(sin⁡2(x))2cos⁡2(x)sin⁡2(x)−6log⁡(sin⁡2(x))−6log⁡(sin⁡2(x))cos⁡2(x)sin⁡2(x)+4cos⁡2(x)sin⁡2(x))cos⁡(x)log⁡(2)3sin⁡(x)\frac{12 \left(\log{\left(\sin^{2}{\left(x \right)} \right)}^{2} + \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 6 \log{\left(\sin^{2}{\left(x \right)} \right)} - \frac{6 \log{\left(\sin^{2}{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\log{\left(2 \right)}^{3} \sin{\left(x \right)}}
Simplify
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