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Derivative of:
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Identical expressions
cos(2x)/(cos(x)-sin(x))
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Similar expressions
cos(2x)/(cos(x)+sin(x))
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The derivative of the function/ cos(2x)/(cos(x)-sin(x))

Derivative of cos(2x)/(cos(x)-sin(x))

The solution

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    cos(2*x)   
---------------
cos(x) - sin(x)

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

cos(2*x)/(cos(x) - sin(x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(2 x \right)}$ and $g{\left(x \right)} = - \sin{\left(x \right)} + \cos{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 x$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
  1. 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 of the chain rule is:
  
  $- 2 \sin{\left(2 x \right)}$
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 sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    So, the result is: $- \cos{\left(x \right)}$
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result is: $- \sin{\left(x \right)} - \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

             /                       2\                                        
             |    2*(cos(x) + sin(x)) |            4*(cos(x) + sin(x))*sin(2*x)
4*cos(2*x) - |1 + --------------------|*cos(2*x) - ----------------------------
             |                      2 |                  -cos(x) + sin(x)      
             \    (-cos(x) + sin(x))  /                                        
-------------------------------------------------------------------------------
                                -cos(x) + sin(x)

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

Simplify

The third derivative [src]

                                                                                      /                       2\                           
                                                                                      |    6*(cos(x) + sin(x)) |                           
                                                                                      |5 + --------------------|*(cos(x) + sin(x))*cos(2*x)
                /                       2\                                            |                      2 |                           
                |    2*(cos(x) + sin(x)) |            12*(cos(x) + sin(x))*cos(2*x)   \    (-cos(x) + sin(x))  /                           
-8*sin(2*x) + 6*|1 + --------------------|*sin(2*x) - ----------------------------- + -----------------------------------------------------
                |                      2 |                   -cos(x) + sin(x)                            -cos(x) + sin(x)                  
                \    (-cos(x) + sin(x))  /                                                                                                 
-------------------------------------------------------------------------------------------------------------------------------------------
                                                              -cos(x) + sin(x)

$$\frac{6 \left(1 + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \sin{\left(2 x \right)} + \frac{\left(5 + \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(2 x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} - 8 \sin{\left(2 x \right)} - \frac{12 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos{\left(2 x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$

Simplify

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Derivative of cos(2x)/(cos(x)-sin(x))

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The solution