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The derivative of the function/ cot(3*x-1)/x

Derivative of cot(3*x-1)/x

The solution

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

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

cot(3*x - 1)/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \cot{\left(3 x - 1 \right)}$ and $g{\left(x \right)} = x$ .

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

$\frac{- \frac{x \left(3 \sin^{2}{\left(3 x - 1 \right)} + 3 \cos^{2}{\left(3 x - 1 \right)}\right)}{\cos^{2}{\left(3 x - 1 \right)} \tan^{2}{\left(3 x - 1 \right)}} - \cot{\left(3 x - 1 \right)}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of cot(3*x-1)/x

The graph:

Piecewise:

The solution

Method #1

Method #2