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Identical expressions
one /(seven *ctg^2x)
1 divide by (7 multiply by ctg squared x)
one divide by (seven multiply by ctg squared x)
1/(7*ctg2x)
1/7*ctg2x
1/(7*ctg²x)
1/(7*ctg to the power of 2x)
1/(7ctg^2x)
1/(7ctg2x)
1/7ctg2x
1/7ctg^2x
1 divide by (7*ctg^2x)

The derivative of the function/ 1/(7*ctg^2x)

Derivative of 1/(7*ctg^2x)

The solution

You have entered [src]

      1    
1*---------
       2   
  7*cot (x)

$$1 \cdot \frac{1}{7 \cot^{2}{\left(x \right)}}$$

d /      1    \
--|1*---------|
dx|       2   |
  \  7*cot (x)/

$$\frac{d}{d x} 1 \cdot \frac{1}{7 \cot^{2}{\left(x \right)}}$$

Transform

Detail solution

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 $g{\left(x \right)} = 7 \cot^{2}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cot{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(x \right)}$ :
    1. There are multiple ways to do this derivative.
      Method #1
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}$
      
      Let $u = \tan{\left(x \right)}$ .
      
      Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      Rewrite the function to be differentiated:
      
      $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
      
      The result of the chain rule is:
      
      $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
      Method #2
      
      Rewrite the function to be differentiated:
      
      $\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
      
      Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    The result of the chain rule is:
    
    $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  So, the result is: $- \frac{14 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
Now plug in to the quotient rule:

$\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{7 \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)}}$
Now simplify:

$\frac{2 \tan{\left(x \right)}}{7 \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{2 \tan{\left(x \right)}}{7 \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1     /          2   \ 
- ---------*\-2 - 2*cot (x)/ 
       2                     
  7*cot (x)                  
-----------------------------
            cot(x)

$$- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \frac{1}{7 \cot^{2}{\left(x \right)}}}{\cot{\left(x \right)}}$$

Simplify

The second derivative [src]

                /       /       2   \\
  /       2   \ |     3*\1 + cot (x)/|
2*\1 + cot (x)/*|-2 + ---------------|
                |            2       |
                \         cot (x)    /
--------------------------------------
                   2                  
              7*cot (x)

$$\frac{2 \cdot \left(\frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{2}{\left(x \right)}} - 2\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{7 \cot^{2}{\left(x \right)}}$$

Simplify

The third derivative [src]

                /                                     2\
                |      /       2   \     /       2   \ |
  /       2   \ |    4*\1 + cot (x)/   3*\1 + cot (x)/ |
8*\1 + cot (x)/*|1 - --------------- + ----------------|
                |           2                 4        |
                \        cot (x)           cot (x)     /
--------------------------------------------------------
                        7*cot(x)

$$\frac{8 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{4}{\left(x \right)}} - \frac{4 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{2}{\left(x \right)}} + 1\right)}{7 \cot{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of 1/(7*ctg^2x)

The graph:

Piecewise:

The solution

Method #1

Method #2