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Identical expressions
(log two (sinx^2))^ three
( logarithm of 2( sinus of x squared )) cubed
( logarithm of two ( sinus of x squared )) to the power of three
(log2(sinx2))3
log2sinx23
(log2(sinx²))³
(log2(sinx to the power of 2)) to the power of 3
log2sinx^2^3

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

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

The solution

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              3
/   /   2   \\ 
|log\sin (x)/| 
|------------| 
\   log(2)   /

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

(log(sin(x)^2)/log(2))^3

Detail solution

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

$\frac{6 \log{\left(\sin^{2}{\left(x \right)} \right)}^{2} \cos{\left(x \right)}}{\log{\left(2 \right)}^{3} \sin{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

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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)

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

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)

$$\frac{6 \left(- \log{\left(\sin^{2}{\left(x \right)} \right)} - \frac{\log{\left(\sin^{2}{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\sin^{2}{\left(x \right)} \right)}}{\log{\left(2 \right)}^{3}}$$

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)

$$\frac{12 \left(\log{\left(\sin^{2}{\left(x \right)} \right)}^{2} + \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}^{2} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - 6 \log{\left(\sin^{2}{\left(x \right)} \right)} - \frac{6 \log{\left(\sin^{2}{\left(x \right)} \right)} \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{4 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\log{\left(2 \right)}^{3} \sin{\left(x \right)}}$$

Simplify

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

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The solution