Derivative of tgx/(sinx-cosx)

The solution

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     tan(x)    
---------------
sin(x) - cos(x)

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

tan(x)/(sin(x) - cos(x))

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sin{\left(x \right)} - \cos{\left(x \right)}$ term by term:
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $\sin{\left(x \right)}$
  2. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $\sin{\left(x \right)} + \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2                                 
  1 + tan (x)     (-cos(x) - sin(x))*tan(x)
--------------- + -------------------------
sin(x) - cos(x)                        2   
                      (sin(x) - cos(x))

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                               /                       2\                                                                    
                                                                               |    6*(cos(x) + sin(x)) |                                                                    
                                                                               |5 + --------------------|*(cos(x) + sin(x))*tan(x)                                           
                                                  /                       2\   |                      2 |                              /       2   \                         
  /       2   \ /         2   \     /       2   \ |    2*(cos(x) + sin(x)) |   \    (-cos(x) + sin(x))  /                            6*\1 + tan (x)/*(cos(x) + sin(x))*tan(x)
2*\1 + tan (x)/*\1 + 3*tan (x)/ + 3*\1 + tan (x)/*|1 + --------------------| - --------------------------------------------------- - ----------------------------------------
                                                  |                      2 |                     -cos(x) + sin(x)                                -cos(x) + sin(x)            
                                                  \    (-cos(x) + sin(x))  /                                                                                                 
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               -cos(x) + sin(x)

$$\frac{3 \left(1 + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\left(5 + \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \tan{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$

Simplify

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Derivative of tgx/(sinx-cosx)

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