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Derivative of:
Derivative of 5*x
Derivative of (1-x^2)^(1/2)
Derivative of x^(3*x)
Derivative of (x+1)/(x-1)
Identical expressions
ln(ctg(x^ one / three))
ln(ctg(x to the power of 1 divide by 3))
ln(ctg(x to the power of one divide by three))
ln(ctg(x1/3))
lnctgx1/3
lnctgx^1/3
ln(ctg(x^1 divide by 3))

The derivative of the function/ ln(ctg(x^1/3))

Derivative of ln(ctg(x^1/3))

The solution

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   /   /3 ___\\
log\cot\\/ x //

$$\log{\left(\cot{\left(\sqrt[3]{x} \right)} \right)}$$

d /   /   /3 ___\\\
--\log\cot\\/ x ///
dx

$$\frac{d}{d x} \log{\left(\cot{\left(\sqrt[3]{x} \right)} \right)}$$

Transform

Detail solution

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

$- \frac{\frac{\sin^{2}{\left(\sqrt[3]{x} \right)}}{3 x^{\frac{2}{3}}} + \frac{\cos^{2}{\left(\sqrt[3]{x} \right)}}{3 x^{\frac{2}{3}}}}{\cos^{2}{\left(\sqrt[3]{x} \right)} \tan^{2}{\left(\sqrt[3]{x} \right)} \cot{\left(\sqrt[3]{x} \right)}}$
Now simplify:

$- \frac{2}{3 x^{\frac{2}{3}} \sin{\left(2 \sqrt[3]{x} \right)}}$

The answer is:

$- \frac{2}{3 x^{\frac{2}{3}} \sin{\left(2 \sqrt[3]{x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2/3 ___\
 -1 - cot \\/ x /
-----------------
   2/3    /3 ___\
3*x   *cot\\/ x /

$$\frac{- \cot^{2}{\left(\sqrt[3]{x} \right)} - 1}{3 x^{\frac{2}{3}} \cot{\left(\sqrt[3]{x} \right)}}$$

Simplify

The second derivative [src]

                  /           2/3 ___\                   \
/       2/3 ___\\ |    1 + cot \\/ x /          2        |
\1 + cot \\/ x //*|2 - --------------- + ----------------|
                  |         2/3 ___\     3 ___    /3 ___\|
                  \      cot \\/ x /     \/ x *cot\\/ x //
----------------------------------------------------------
                             4/3                          
                          9*x

$$\frac{\left(\cot^{2}{\left(\sqrt[3]{x} \right)} + 1\right) \left(- \frac{\cot^{2}{\left(\sqrt[3]{x} \right)} + 1}{\cot^{2}{\left(\sqrt[3]{x} \right)}} + 2 + \frac{2}{\sqrt[3]{x} \cot{\left(\sqrt[3]{x} \right)}}\right)}{9 x^{\frac{4}{3}}}$$

Simplify

The third derivative [src]

                    /                                                           2                                            \
                    |                                /3 ___\   /       2/3 ___\\      /       2/3 ___\\     /       2/3 ___\\|
  /       2/3 ___\\ |   6            5          2*cot\\/ x /   \1 + cot \\/ x //    2*\1 + cot \\/ x //   3*\1 + cot \\/ x //|
2*\1 + cot \\/ x //*|- ---- - --------------- - ------------ - ------------------ + ------------------- + -------------------|
                    |   7/3    8/3    /3 ___\         2           2    3/3 ___\         2    /3 ___\         7/3    2/3 ___\ |
                    \  x      x   *cot\\/ x /        x           x *cot \\/ x /        x *cot\\/ x /        x   *cot \\/ x / /
------------------------------------------------------------------------------------------------------------------------------
                                                              27

$$\frac{2 \left(\cot^{2}{\left(\sqrt[3]{x} \right)} + 1\right) \left(- \frac{\left(\cot^{2}{\left(\sqrt[3]{x} \right)} + 1\right)^{2}}{x^{2} \cot^{3}{\left(\sqrt[3]{x} \right)}} + \frac{2 \left(\cot^{2}{\left(\sqrt[3]{x} \right)} + 1\right)}{x^{2} \cot{\left(\sqrt[3]{x} \right)}} - \frac{2 \cot{\left(\sqrt[3]{x} \right)}}{x^{2}} + \frac{3 \left(\cot^{2}{\left(\sqrt[3]{x} \right)} + 1\right)}{x^{\frac{7}{3}} \cot^{2}{\left(\sqrt[3]{x} \right)}} - \frac{6}{x^{\frac{7}{3}}} - \frac{5}{x^{\frac{8}{3}} \cot{\left(\sqrt[3]{x} \right)}}\right)}{27}$$

Simplify

The graph

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Identical expressions

Derivative of ln(ctg(x^1/3))

The graph:

Piecewise:

The solution

Method #1

Method #2