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y=1/ctg^2x-1/3(ctgx)

The derivative of the function/ y=1/ctg^2x+1/3(ctgx)

Derivative of y=1/ctg^2x+1/3(ctgx)

The solution

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     1      cot(x)
1*------- + ------
     2        3   
  cot (x)

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

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

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

Transform

Detail solution

Differentiate $\frac{\cot{\left(x \right)}}{3} + 1 \cdot \frac{1}{\cot^{2}{\left(x \right)}}$ term by term:
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 $g{\left(x \right)} = \cot^{2}{\left(x \right)}$ .
  
  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. 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:
    
    $- \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cot{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  Now plug in to the quotient rule:
  
  $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. $\frac{d}{d x} \cot{\left(x \right)} = - \frac{1}{\sin^{2}{\left(x \right)}}$
  So, the result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
The result is: $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{3 \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2                2   
  1   cot (x)   -2 - 2*cot (x)
- - - ------- - --------------
  3      3                2   
                cot(x)*cot (x)

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

Simplify

The second derivative [src]

  /       2   \ /              /       2   \         \
  |1   cot (x)| |     6      9*\1 + cot (x)/         |
2*|- + -------|*|- ------- + --------------- + cot(x)|
  \3      3   / |     2             4                |
                \  cot (x)       cot (x)             /

$$2 \left(\frac{\cot^{2}{\left(x \right)}}{3} + \frac{1}{3}\right) \left(\frac{9 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{4}{\left(x \right)}} + \cot{\left(x \right)} - \frac{6}{\cot^{2}{\left(x \right)}}\right)$$

Simplify

The third derivative [src]

                /                                                             2\
  /       2   \ |                             /       2   \      /       2   \ |
  |1   cot (x)| |          2        12     48*\1 + cot (x)/   36*\1 + cot (x)/ |
2*|- + -------|*|-1 - 3*cot (x) + ------ - ---------------- + -----------------|
  \3      3   / |                 cot(x)          3                   5        |
                \                              cot (x)             cot (x)     /

$$2 \left(\frac{\cot^{2}{\left(x \right)}}{3} + \frac{1}{3}\right) \left(\frac{36 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{5}{\left(x \right)}} - \frac{48 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{3}{\left(x \right)}} - 3 \cot^{2}{\left(x \right)} - 1 + \frac{12}{\cot{\left(x \right)}}\right)$$

Simplify

The graph

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Derivative of y=1/ctg^2x+1/3(ctgx)

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Piecewise:

The solution

Method #1

Method #2