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Derivative of:
Derivative of sin(4x)
Derivative of y=(x^4)÷2
Derivative of (x^2-6x-33)e^(2x-11)
Derivative of ln(x+10)^11
Identical expressions
ctg(lnx)*sin^ three (x)
ctg(lnx) multiply by sinus of cubed (x)
ctg(lnx) multiply by sinus of to the power of three (x)
ctg(lnx)*sin3(x)
ctglnx*sin3x
ctg(lnx)*sin³(x)
ctg(lnx)*sin to the power of 3(x)
ctg(lnx)sin^3(x)
ctg(lnx)sin3(x)
ctglnxsin3x
ctglnxsin^3x

The derivative of the function/ ctg(lnx)*sin^3(x)

Derivative of ctg(lnx)*sin^3(x)

The solution

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               3   
cot(log(x))*sin (x)

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

cot(log(x))*sin(x)^3

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \cot{\left(\log{\left(x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\log{\left(x \right)} \right)} = \frac{1}{\tan{\left(\log{\left(x \right)} \right)}}$
  2. Let $u = \tan{\left(\log{\left(x \right)} \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\log{\left(x \right)} \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(\log{\left(x \right)} \right)} = \frac{\sin{\left(\log{\left(x \right)} \right)}}{\cos{\left(\log{\left(x \right)} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\log{\left(x \right)} \right)}$ and $g{\left(x \right)} = \cos{\left(\log{\left(x \right)} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \log{\left(x \right)}$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      
      The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \log{\left(x \right)}$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      
      The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
      
      Now plug in to the quotient rule:
      
      $\frac{\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\cos^{2}{\left(\log{\left(x \right)} \right)}}$
    The result of the chain rule is:
    
    $- \frac{\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\cos^{2}{\left(\log{\left(x \right)} \right)} \tan^{2}{\left(\log{\left(x \right)} \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\log{\left(x \right)} \right)} = \frac{\cos{\left(\log{\left(x \right)} \right)}}{\sin{\left(\log{\left(x \right)} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(\log{\left(x \right)} \right)}$ and $g{\left(x \right)} = \sin{\left(\log{\left(x \right)} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    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 $\frac{d}{d x} \log{\left(x \right)}$ :
      
      The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      
      The result of the chain rule is:
      
      $- \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \log{\left(x \right)}$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      
      The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      
      The result of the chain rule is:
      
      $\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}$
    Now plug in to the quotient rule:
    
    $\frac{- \frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} - \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}}{\sin^{2}{\left(\log{\left(x \right)} \right)}}$
$g{\left(x \right)} = \sin^{3}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  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:
  
  $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
The result is: $- \frac{\left(\frac{\sin^{2}{\left(\log{\left(x \right)} \right)}}{x} + \frac{\cos^{2}{\left(\log{\left(x \right)} \right)}}{x}\right) \sin^{3}{\left(x \right)}}{\cos^{2}{\left(\log{\left(x \right)} \right)} \tan^{2}{\left(\log{\left(x \right)} \right)}} + 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} \cot{\left(\log{\left(x \right)} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3    /        2        \                               
sin (x)*\-1 - cot (log(x))/        2                      
--------------------------- + 3*sin (x)*cos(x)*cot(log(x))
             x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                        3    /       2        \ /         2                        \     /       2        \ /   2           2   \               2    /       2        \                           
    /       2           2   \                      2*sin (x)*\1 + cot (log(x))/*\2 + 3*cot (log(x)) + 3*cot(log(x))/   9*\1 + cot (log(x))/*\sin (x) - 2*cos (x)/*sin(x)   9*sin (x)*\1 + cot (log(x))/*(1 + 2*cot(log(x)))*cos(x)
- 3*\- 2*cos (x) + 7*sin (x)/*cos(x)*cot(log(x)) - ----------------------------------------------------------------- + ------------------------------------------------- + -------------------------------------------------------
                                                                                    3                                                          x                                                       2                          
                                                                                   x                                                                                                                  x

$$- 3 \left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} \cot{\left(\log{\left(x \right)} \right)} + \frac{9 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \sin{\left(x \right)}}{x} + \frac{9 \left(2 \cot{\left(\log{\left(x \right)} \right)} + 1\right) \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \left(\cot^{2}{\left(\log{\left(x \right)} \right)} + 1\right) \left(3 \cot^{2}{\left(\log{\left(x \right)} \right)} + 3 \cot{\left(\log{\left(x \right)} \right)} + 2\right) \sin^{3}{\left(x \right)}}{x^{3}}$$

Simplify

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Derivative of ctg(lnx)*sin^3(x)

The graph:

Piecewise:

The solution

Method #1

Method #2