Derivative of (tan(x)-cot(x))/(tan(x)+cot(x))

The solution

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tan(x) - cot(x)
---------------
tan(x) + cot(x)

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

(tan(x) - cot(x))/(tan(x) + cot(x))

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\tan{\left(x \right)} - \cot{\left(x \right)}$ term by term:
  1. 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 $\frac{d}{d x} \tan{\left(x \right)}$ :
      
      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:
      
      $\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
    So, the result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
  2. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\tan{\left(x \right)} + \cot{\left(x \right)}$ term by term:
  1. $\frac{d}{d x} \cot{\left(x \right)} = - \frac{1}{\sin^{2}{\left(x \right)}}$
  2. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
  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 plug in to the quotient rule:

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

$\frac{4}{\tan{\left(x \right)}} - \frac{4 \cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}$

The answer is:

$\frac{4}{\tan{\left(x \right)}} - \frac{4 \cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of (tan(x)-cot(x))/(tan(x)+cot(x))

Other calculators

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Identical expressions

Similar expressions

Derivative of (tan(x)-cot(x))/(tan(x)+cot(x))

The graph:

Piecewise:

The solution

Method #1

Method #2