Derivative of (tgx+1)/(tgx-1)

The solution

You have entered [src]

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

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

d /tan(x) + 1\
--|----------|
dx\tan(x) - 1/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\tan{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  3. 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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\tan{\left(x \right)} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. $\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (tgx+1)/(tgx-1)

The graph:

Piecewise:

The solution