Derivative of tgx-(1/tgx)

The solution

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           1   
tan(x) - ------
         tan(x)

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

tan(x) - 1/tan(x)

Detail solution

Differentiate $\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}$ 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. Let $u = \tan{\left(x \right)}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
    1. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
    The result of the chain rule is:
    
    $- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
Now simplify:

$\frac{8}{1 - \cos{\left(4 x \right)}}$

The answer is:

$\frac{8}{1 - \cos{\left(4 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      2   
       2      -1 - tan (x)
1 + tan (x) - ------------
                   2      
                tan (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                2                                            3\
  |                 2                 /       2   \                                /       2   \ |
  |    /       2   \         2      5*\1 + tan (x)/         2    /       2   \   3*\1 + tan (x)/ |
2*|2 + \1 + tan (x)/  + 2*tan (x) - ---------------- + 2*tan (x)*\1 + tan (x)/ + ----------------|
  |                                        2                                            4        |
  \                                     tan (x)                                      tan (x)     /

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

Simplify

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Derivative of tgx-(1/tgx)

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Piecewise:

The solution