Derivative of tgx-ctgx-2x

The solution

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tan(x) - cot(x) - 2*x

$$- 2 x + \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)$$

tan(x) - cot(x) - 2*x

Detail solution

Differentiate $- 2 x + \left(\tan{\left(x \right)} - \cot{\left(x \right)}\right)$ term by term:
1. Differentiate $\tan{\left(x \right)} - \cot{\left(x \right)}$ term by term:
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. 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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    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 $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
      
      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:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    So, the result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $-2$
The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - 2$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2         2   
cot (x) + tan (x)

$$\tan^{2}{\left(x \right)} + \cot^{2}{\left(x \right)}$$

Simplify

The second derivative [src]

  //       2   \          /       2   \       \
2*\\1 + tan (x)/*tan(x) - \1 + cot (x)/*cot(x)/

$$2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)$$

Simplify

The third derivative [src]

  /             2                2                                                    \
  |/       2   \    /       2   \         2    /       2   \        2    /       2   \|
2*\\1 + cot (x)/  + \1 + tan (x)/  + 2*cot (x)*\1 + cot (x)/ + 2*tan (x)*\1 + tan (x)//

$$2 \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)}\right)$$

Simplify

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Derivative of tgx-ctgx-2x

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Piecewise:

The solution

Method #1

Method #2