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y=(ln^3sinx)/cos2x
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y=(ln3sinx)/cos2x
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y=ln^3sinx/cos2x
y=(ln^3sinx) divide by cos2x

The derivative of the function/ y=(ln^3sinx)/cos2x

Derivative of y=(ln^3sinx)/cos2x

The solution

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   3          
log (x)*sin(x)
--------------
   cos(2*x)

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

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

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)}^{3} \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(2 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(x \right)}^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  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:
    
    $\frac{3 \log{\left(x \right)}^{2}}{x}$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $\log{\left(x \right)}^{3} \cos{\left(x \right)} + \frac{3 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 2 x$ .
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 $\frac{d}{d x} 2 x$ :
  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:
  
  $- 2 \sin{\left(2 x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/                                                                                                     /                3*sin(x)\                \       
|                                                      /         2     \                            4*|cos(x)*log(x) + --------|*log(x)*sin(2*x)|       
|     2             3*(-2 + log(x))*sin(x)        2    |    2*sin (2*x)|          6*cos(x)*log(x)     \                   x    /                |       
|- log (x)*sin(x) - ---------------------- + 4*log (x)*|1 + -----------|*sin(x) + --------------- + --------------------------------------------|*log(x)
|                              2                       |        2      |                 x                            cos(2*x)                  |       
\                             x                        \     cos (2*x) /                                                                        /       
--------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        cos(2*x)

$$\frac{\left(\frac{4 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x}\right) \log{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} + 4 \left(\frac{2 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \log{\left(x \right)}^{2} \sin{\left(x \right)} - \log{\left(x \right)}^{2} \sin{\left(x \right)} + \frac{6 \log{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)}}{\cos{\left(2 x \right)}}$$

Simplify

The third derivative [src]

                                                                                                                                                                                                                                                              /         2     \                
                                                                                                                                                                      /   2             6*cos(x)*log(x)   3*(-2 + log(x))*sin(x)\                        3    |    6*sin (2*x)|                
                                                                                                                                                                    6*|log (x)*sin(x) - --------------- + ----------------------|*log(x)*sin(2*x)   8*log (x)*|5 + -----------|*sin(x)*sin(2*x)
                        2               /       2              \                     /         2     \                                                                |                        x                     2          |                             |        2      |                
     3             9*log (x)*sin(x)   6*\1 + log (x) - 3*log(x)/*sin(x)         2    |    2*sin (2*x)| /                3*sin(x)\   9*(-2 + log(x))*cos(x)*log(x)     \                                             x           /                             \     cos (2*x) /                
- log (x)*cos(x) - ---------------- + --------------------------------- + 12*log (x)*|1 + -----------|*|cos(x)*log(x) + --------| - ----------------------------- - ----------------------------------------------------------------------------- + -------------------------------------------
                          x                            3                             |        2      | \                   x    /                  2                                                   cos(2*x)                                                       cos(2*x)                 
                                                      x                              \     cos (2*x) /                                            x                                                                                                                                            
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                            cos(2*x)

$$\frac{12 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x}\right) \left(\frac{2 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \log{\left(x \right)}^{2} + \frac{8 \left(\frac{6 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 5\right) \log{\left(x \right)}^{3} \sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} - \frac{6 \left(\log{\left(x \right)}^{2} \sin{\left(x \right)} - \frac{6 \log{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \left(\log{\left(x \right)} - 2\right) \sin{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} - \log{\left(x \right)}^{3} \cos{\left(x \right)} - \frac{9 \log{\left(x \right)}^{2} \sin{\left(x \right)}}{x} - \frac{9 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} \cos{\left(x \right)}}{x^{2}} + \frac{6 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right) \sin{\left(x \right)}}{x^{3}}}{\cos{\left(2 x \right)}}$$

Simplify

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Identical expressions

Derivative of y=(ln^3sinx)/cos2x

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The solution