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Derivative of:
Derivative of 6
Derivative of e^(x*(-3))
Derivative of x-cos(x)
Derivative of x*cos(2*x)
Identical expressions
ctg^ eight (x^ one / two +x^ one / five)
ctg to the power of 8(x to the power of 1 divide by 2 plus x to the power of 1 divide by 5)
ctg to the power of eight (x to the power of one divide by two plus x to the power of one divide by five)
ctg8(x1/2+x1/5)
ctg8x1/2+x1/5
ctg⁸(x^1/2+x^1/5)
ctg^8x^1/2+x^1/5
ctg^8(x^1 divide by 2+x^1 divide by 5)
Similar expressions
ctg^8(x^1/2-x^1/5)

The derivative of the function/ ctg^8(x^1/2+x^1/5)

Derivative of ctg^8(x^1/2+x^1/5)

The solution

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   8/  ___   5 ___\
cot \\/ x  + \/ x /

$$\cot^{8}{\left(\sqrt[5]{x} + \sqrt{x} \right)}$$

d /   8/  ___   5 ___\\
--\cot \\/ x  + \/ x //
dx

$$\frac{d}{d x} \cot^{8}{\left(\sqrt[5]{x} + \sqrt{x} \right)}$$

Transform

Detail solution

Let $u = \cot{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ .
Apply the power rule: $u^{8}$ goes to $8 u^{7}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\sqrt[5]{x} + \sqrt{x} \right)} = \frac{1}{\tan{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
  2. Let $u = \tan{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(\sqrt[5]{x} + \sqrt{x} \right)} = \frac{\sin{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{\cos{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ and $g{\left(x \right)} = \cos{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = \sqrt[5]{x} + \sqrt{x}$ .
      
      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 $\frac{d}{d x} \left(\sqrt[5]{x} + \sqrt{x}\right)$ :
      
      Differentiate $\sqrt[5]{x} + \sqrt{x}$ term by term:
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      Apply the power rule: $\sqrt[5]{x}$ goes to $\frac{1}{5 x^{\frac{4}{5}}}$
      
      The result is: $\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}$
      
      The result of the chain rule is:
      
      $\left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \cos{\left(\sqrt[5]{x} + \sqrt{x} \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = \sqrt[5]{x} + \sqrt{x}$ .
      
      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 $\frac{d}{d x} \left(\sqrt[5]{x} + \sqrt{x}\right)$ :
      
      Differentiate $\sqrt[5]{x} + \sqrt{x}$ term by term:
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      Apply the power rule: $\sqrt[5]{x}$ goes to $\frac{1}{5 x^{\frac{4}{5}}}$
      
      The result is: $\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}$
      
      The result of the chain rule is:
      
      $- \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \sin{\left(\sqrt[5]{x} + \sqrt{x} \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{\left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \sin^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{\cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
    The result of the chain rule is:
    
    $- \frac{\left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \sin^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{\cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} \tan^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(\sqrt[5]{x} + \sqrt{x} \right)} = \frac{\cos{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{\sin{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ and $g{\left(x \right)} = \sin{\left(\sqrt[5]{x} + \sqrt{x} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \sqrt[5]{x} + \sqrt{x}$ .
    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 $\frac{d}{d x} \left(\sqrt[5]{x} + \sqrt{x}\right)$ :
      
      Differentiate $\sqrt[5]{x} + \sqrt{x}$ term by term:
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      Apply the power rule: $\sqrt[5]{x}$ goes to $\frac{1}{5 x^{\frac{4}{5}}}$
      
      The result is: $\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}$
      
      The result of the chain rule is:
      
      $- \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \sin{\left(\sqrt[5]{x} + \sqrt{x} \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = \sqrt[5]{x} + \sqrt{x}$ .
    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 $\frac{d}{d x} \left(\sqrt[5]{x} + \sqrt{x}\right)$ :
      
      Differentiate $\sqrt[5]{x} + \sqrt{x}$ term by term:
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      Apply the power rule: $\sqrt[5]{x}$ goes to $\frac{1}{5 x^{\frac{4}{5}}}$
      
      The result is: $\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}$
      
      The result of the chain rule is:
      
      $\left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \cos{\left(\sqrt[5]{x} + \sqrt{x} \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \sin^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} - \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{\sin^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
The result of the chain rule is:

$- \frac{8 \left(\left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \sin^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}\right) \cot^{7}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{\cos^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} \tan^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$
Now simplify:

$\frac{\left(- 4 x^{\frac{4}{5}} - \frac{8 \sqrt{x}}{5}\right) \cos^{7}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{x^{\frac{13}{10}} \sin^{9}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$

The answer is:

$\frac{\left(- 4 x^{\frac{4}{5}} - \frac{8 \sqrt{x}}{5}\right) \cos^{7}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{x^{\frac{13}{10}} \sin^{9}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     7/  ___   5 ___\ /        2/  ___   5 ___\\ /   1        1   \
8*cot \\/ x  + \/ x /*\-1 - cot \\/ x  + \/ x //*|------- + ------|
                                                 |    ___      4/5|
                                                 \2*\/ x    5*x   /

$$8 \cdot \left(\frac{1}{2 \sqrt{x}} + \frac{1}{5 x^{\frac{4}{5}}}\right) \left(- \cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} - 1\right) \cot^{7}{\left(\sqrt[5]{x} + \sqrt{x} \right)}$$

Simplify

The second derivative [src]

                                                /                                                   2                                       2                          \
     6/  ___   5 ___\ /       2/  ___   5 ___\\ |/ 16     25 \    /  ___   5 ___\     / 2       5  \     2/  ___   5 ___\     / 2       5  \  /       2/  ___   5 ___\\|
2*cot \\/ x  + \/ x /*\1 + cot \\/ x  + \/ x //*||---- + ----|*cot\\/ x  + \/ x / + 2*|---- + -----| *cot \\/ x  + \/ x / + 7*|---- + -----| *\1 + cot \\/ x  + \/ x //|
                                                || 9/5    3/2|                        | 4/5     ___|                          | 4/5     ___|                           |
                                                \\x      x   /                        \x      \/ x /                          \x      \/ x /                           /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                   25

$$\frac{2 \left(\cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 1\right) \left(2 \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right)^{2} \cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 7 \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right)^{2} \left(\cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 1\right) + \left(\frac{25}{x^{\frac{3}{2}}} + \frac{16}{x^{\frac{9}{5}}}\right) \cot{\left(\sqrt[5]{x} + \sqrt{x} \right)}\right) \cot^{6}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{25}$$

Simplify

The third derivative [src]

                                               /                                                       3                                                   2               3                                                                         3                                                                                                                             \ 
    5/  ___   5 ___\ /       2/  ___   5 ___\\ |     2/  ___   5 ___\ /  96    125 \     / 2       5  \     4/  ___   5 ___\      /       2/  ___   5 ___\\  / 2       5  \         3/  ___   5 ___\ / 2       5  \ / 16     25 \      / 2       5  \     2/  ___   5 ___\ /       2/  ___   5 ___\\      /       2/  ___   5 ___\\ / 2       5  \ / 16     25 \    /  ___   5 ___\| 
-cot \\/ x  + \/ x /*\1 + cot \\/ x  + \/ x //*|3*cot \\/ x  + \/ x /*|----- + ----| + 4*|---- + -----| *cot \\/ x  + \/ x / + 42*\1 + cot \\/ x  + \/ x // *|---- + -----|  + 6*cot \\/ x  + \/ x /*|---- + -----|*|---- + ----| + 44*|---- + -----| *cot \\/ x  + \/ x /*\1 + cot \\/ x  + \/ x // + 21*\1 + cot \\/ x  + \/ x //*|---- + -----|*|---- + ----|*cot\\/ x  + \/ x /| 
                                               |                      | 14/5    5/2|     | 4/5     ___|                                                      | 4/5     ___|                          | 4/5     ___| | 9/5    3/2|      | 4/5     ___|                                                                               | 4/5     ___| | 9/5    3/2|                   | 
                                               \                      \x       x   /     \x      \/ x /                                                      \x      \/ x /                          \x      \/ x / \x      x   /      \x      \/ x /                                                                               \x      \/ x / \x      x   /                   / 
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                                                                                                                                                                                         125

$$- \frac{\left(\cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 1\right) \left(4 \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right)^{3} \cot^{4}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 44 \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right)^{3} \left(\cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 1\right) \cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 6 \cdot \left(\frac{25}{x^{\frac{3}{2}}} + \frac{16}{x^{\frac{9}{5}}}\right) \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right) \cot^{3}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 42 \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right)^{3} \left(\cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 1\right)^{2} + 21 \cdot \left(\frac{25}{x^{\frac{3}{2}}} + \frac{16}{x^{\frac{9}{5}}}\right) \left(\frac{5}{\sqrt{x}} + \frac{2}{x^{\frac{4}{5}}}\right) \left(\cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 1\right) \cot{\left(\sqrt[5]{x} + \sqrt{x} \right)} + 3 \cdot \left(\frac{125}{x^{\frac{5}{2}}} + \frac{96}{x^{\frac{14}{5}}}\right) \cot^{2}{\left(\sqrt[5]{x} + \sqrt{x} \right)}\right) \cot^{5}{\left(\sqrt[5]{x} + \sqrt{x} \right)}}{125}$$

Simplify

The graph

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Derivative of ctg^8(x^1/2+x^1/5)

The graph:

Piecewise:

The solution

Method #1

Method #2