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Derivative of:
Derivative of -x^4
Derivative of x^2*cos(2*x)
Derivative of y=1/2sin2x
Derivative of x^3/(2(x+1)^2)
Identical expressions
cot(one /(x^ two))
cotangent of (1 divide by (x squared ))
cotangent of (one divide by (x to the power of two))
cot(1/(x2))
cot1/x2
cot(1/(x²))
cot(1/(x to the power of 2))
cot1/x^2
cot(1 divide by (x^2))

The derivative of the function/ cot(1/(x^2))

Derivative of cot(1/(x^2))

The solution

You have entered [src]

   /  1 \
cot|1*--|
   |   2|
   \  x /

$$\cot{\left(1 \cdot \frac{1}{x^{2}} \right)}$$

d /   /  1 \\
--|cot|1*--||
dx|   |   2||
  \   \  x //

$$\frac{d}{d x} \cot{\left(1 \cdot \frac{1}{x^{2}} \right)}$$

Transform

Detail solution

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

$\frac{2}{x^{3} \sin^{2}{\left(\frac{1}{x^{2}} \right)}}$

The answer is:

$\frac{2}{x^{3} \sin^{2}{\left(\frac{1}{x^{2}} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                 /         /1 \     /       2/1 \\        2/1 \\
                 |    9*cot|--|   2*|1 + cot |--||   4*cot |--||
                 |         | 2|     |        | 2||         | 2||
  /       2/1 \\ |         \x /     \        \x //         \x /|
8*|1 + cot |--||*|3 - --------- + ---------------- + ----------|
  |        | 2|| |         2              4               4    |
  \        \x // \        x              x               x     /
----------------------------------------------------------------
                                5                               
                               x

$$\frac{8 \left(\cot^{2}{\left(\frac{1}{x^{2}} \right)} + 1\right) \left(3 - \frac{9 \cot{\left(\frac{1}{x^{2}} \right)}}{x^{2}} + \frac{4 \cot^{2}{\left(\frac{1}{x^{2}} \right)}}{x^{4}} + \frac{2 \left(\cot^{2}{\left(\frac{1}{x^{2}} \right)} + 1\right)}{x^{4}}\right)}{x^{5}}$$

Simplify

The graph

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How to use it?

Identical expressions

Derivative of cot(1/(x^2))

The graph:

Piecewise:

The solution

Method #1

Method #2