Derivative of ctg(√(x+ln(cos(x))))

The solution

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   /  _________________\
cot\\/ x + log(cos(x)) /

$$\cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)}$$

cot(sqrt(x + log(cos(x))))

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                /                                                                                                                                                                     /       2   \                     /       2   \               /       2   \                                       \
                                |                 3                    3                                                  3                                            3                              |    sin (x)| /     sin(x)\       |    sin (x)|               |    sin (x)| /     sin(x)\    /  _________________\|
                                |    /     sin(x)\        /     sin(x)\  /       2/  _________________\\     /     sin(x)\     2/  _________________\     /     sin(x)\     /  _________________\   6*|1 + -------|*|-1 + ------|     8*|1 + -------|*sin(x)     12*|1 + -------|*|-1 + ------|*cot\\/ x + log(cos(x)) /|
/       2/  _________________\\ |  3*|-1 + ------|      2*|-1 + ------| *\1 + cot \\/ x + log(cos(x)) //   4*|-1 + ------| *cot \\/ x + log(cos(x)) /   6*|-1 + ------| *cot\\/ x + log(cos(x)) /     |       2   | \     cos(x)/       |       2   |               |       2   | \     cos(x)/                         |
|1   cot \\/ x + log(cos(x)) /| |    \     cos(x)/        \     cos(x)/                                      \     cos(x)/                                \     cos(x)/                               \    cos (x)/                     \    cos (x)/               \    cos (x)/                                       |
|- + -------------------------|*|-------------------- + ------------------------------------------------ + ------------------------------------------ + ----------------------------------------- + ----------------------------- + -------------------------- + -------------------------------------------------------|
\8               8            / |                 5/2                                  3/2                                             3/2                                           2                                    3/2         _________________                              x + log(cos(x))                    |
                                \(x + log(cos(x)))                    (x + log(cos(x)))                               (x + log(cos(x)))                             (x + log(cos(x)))                    (x + log(cos(x)))          \/ x + log(cos(x)) *cos(x)                                                          /

$$\left(\frac{\cot^{2}{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)}}{8} + \frac{1}{8}\right) \left(\frac{12 \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)}}{x + \log{\left(\cos{\left(x \right)} \right)}} + \frac{6 \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1\right)^{3} \cot{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)}}{\left(x + \log{\left(\cos{\left(x \right)} \right)}\right)^{2}} + \frac{8 \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}{\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \cos{\left(x \right)}} + \frac{2 \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1\right)^{3} \left(\cot^{2}{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)} + 1\right)}{\left(x + \log{\left(\cos{\left(x \right)} \right)}\right)^{\frac{3}{2}}} + \frac{4 \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1\right)^{3} \cot^{2}{\left(\sqrt{x + \log{\left(\cos{\left(x \right)} \right)}} \right)}}{\left(x + \log{\left(\cos{\left(x \right)} \right)}\right)^{\frac{3}{2}}} + \frac{6 \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1\right) \left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)}{\left(x + \log{\left(\cos{\left(x \right)} \right)}\right)^{\frac{3}{2}}} + \frac{3 \left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1\right)^{3}}{\left(x + \log{\left(\cos{\left(x \right)} \right)}\right)^{\frac{5}{2}}}\right)$$

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Derivative of ctg(√(x+ln(cos(x))))

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The solution

Method #1

Method #2