Derivative of cot(x^tan(x))

The solution

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   / tan(x)\
cot\x      /

$$\cot{\left(x^{\tan{\left(x \right)}} \right)}$$

cot(x^tan(x))

Detail solution

There are multiple ways to do this derivative.
Method #1
1. Rewrite the function to be differentiated:
  
  $\cot{\left(x^{\tan{\left(x \right)}} \right)} = \frac{1}{\tan{\left(x^{\tan{\left(x \right)}} \right)}}$
2. Let $u = \tan{\left(x^{\tan{\left(x \right)}} \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x^{\tan{\left(x \right)}} \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(x^{\tan{\left(x \right)}} \right)} = \frac{\sin{\left(x^{\tan{\left(x \right)}} \right)}}{\cos{\left(x^{\tan{\left(x \right)}} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x^{\tan{\left(x \right)}} \right)}$ and $g{\left(x \right)} = \cos{\left(x^{\tan{\left(x \right)}} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = x^{\tan{\left(x \right)}}$ .
    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 $\frac{d}{d x} x^{\tan{\left(x \right)}}$ :
      
      Don't know the steps in finding this derivative.
      
      But the derivative is
      
      $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)}$
      
      The result of the chain rule is:
      
      $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \cos{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = x^{\tan{\left(x \right)}}$ .
    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 $\frac{d}{d x} x^{\tan{\left(x \right)}}$ :
      
      Don't know the steps in finding this derivative.
      
      But the derivative is
      
      $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)}$
      
      The result of the chain rule is:
      
      $- \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \sin{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)} + \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \cos^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}}{\cos^{2}{\left(x^{\tan{\left(x \right)}} \right)}}$
  The result of the chain rule is:
  
  $- \frac{\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)} + \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \cos^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}}{\cos^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{2}{\left(x^{\tan{\left(x \right)}} \right)}}$
Method #2
1. Rewrite the function to be differentiated:
  
  $\cot{\left(x^{\tan{\left(x \right)}} \right)} = \frac{\cos{\left(x^{\tan{\left(x \right)}} \right)}}{\sin{\left(x^{\tan{\left(x \right)}} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \cos{\left(x^{\tan{\left(x \right)}} \right)}$ and $g{\left(x \right)} = \sin{\left(x^{\tan{\left(x \right)}} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = x^{\tan{\left(x \right)}}$ .
  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 $\frac{d}{d x} x^{\tan{\left(x \right)}}$ :
    1. Don't know the steps in finding this derivative.
      
      But the derivative is
      
      $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)}$
    The result of the chain rule is:
    
    $- \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \sin{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x^{\tan{\left(x \right)}}$ .
  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 $\frac{d}{d x} x^{\tan{\left(x \right)}}$ :
    1. Don't know the steps in finding this derivative.
      
      But the derivative is
      
      $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \tan^{\tan{\left(x \right)}}{\left(x \right)}$
    The result of the chain rule is:
    
    $\left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \cos{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)} - \left(\log{\left(\tan{\left(x \right)} \right)} + 1\right) \cos^{2}{\left(x^{\tan{\left(x \right)}} \right)} \tan^{\tan{\left(x \right)}}{\left(x \right)}}{\sin^{2}{\left(x^{\tan{\left(x \right)}} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 tan(x) /        2/ tan(x)\\ /tan(x)   /       2   \       \
x      *\-1 - cot \x      //*|------ + \1 + tan (x)/*log(x)|
                             \  x                          /

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

Simplify

The second derivative [src]

                            /                                 2              /       2   \                                                                            2             \
 tan(x) /       2/ tan(x)\\ |  /tan(x)   /       2   \       \    tan(x)   2*\1 + tan (x)/     /       2   \                    tan(x) /tan(x)   /       2   \       \     / tan(x)\|
x      *\1 + cot \x      //*|- |------ + \1 + tan (x)/*log(x)|  + ------ - --------------- - 2*\1 + tan (x)/*log(x)*tan(x) + 2*x      *|------ + \1 + tan (x)/*log(x)| *cot\x      /|
                            |  \  x                          /       2            x                                                    \  x                          /              |
                            \                                       x                                                                                                               /

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

Simplify

The third derivative [src]

                            /                                 3                                     /             /       2   \                                \                             2            /       2   \     /       2   \                                                     3                                                                                             3                                                                3                                                          /             /       2   \                                \             \
 tan(x) /       2/ tan(x)\\ |  /tan(x)   /       2   \       \      /tan(x)   /       2   \       \ |  tan(x)   2*\1 + tan (x)/     /       2   \              |   2*tan(x)     /       2   \           3*\1 + tan (x)/   6*\1 + tan (x)/*tan(x)      2*tan(x) /tan(x)   /       2   \       \     2/ tan(x)\        2    /       2   \             2*tan(x) /tan(x)   /       2   \       \  /       2/ tan(x)\\      tan(x) /tan(x)   /       2   \       \     / tan(x)\      tan(x) /tan(x)   /       2   \       \ |  tan(x)   2*\1 + tan (x)/     /       2   \              |    / tan(x)\|
x      *\1 + cot \x      //*|- |------ + \1 + tan (x)/*log(x)|  - 3*|------ + \1 + tan (x)/*log(x)|*|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)| - -------- - 2*\1 + tan (x)/ *log(x) + --------------- - ---------------------- - 4*x        *|------ + \1 + tan (x)/*log(x)| *cot \x      / - 4*tan (x)*\1 + tan (x)/*log(x) - 2*x        *|------ + \1 + tan (x)/*log(x)| *\1 + cot \x      // + 6*x      *|------ + \1 + tan (x)/*log(x)| *cot\x      / + 6*x      *|------ + \1 + tan (x)/*log(x)|*|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*cot\x      /|
                            |  \  x                          /      \  x                          / |     2            x                                       |       3                                        2                   x                          \  x                          /                                                               \  x                          /                                  \  x                          /                           \  x                          / |     2            x                                       |             |
                            \                                                                       \    x                                                     /      x                                        x                                                                                                                                                                                                                                                                                                        \    x                                                     /             /

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

Simplify

The graph

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Identical expressions

Derivative of cot(x^tan(x))

The graph:

Piecewise:

The solution

Method #1

Method #2