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(1-cos2x)/cosx

The derivative of the function/ (1+cos2x)/cosx

Derivative of (1+cos2x)/cosx

The solution

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

$$\frac{\cos{\left(2 x \right)} + 1}{\cos{\left(x \right)}}$$

(1 + cos(2*x))/cos(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(2 x \right)} + 1$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\cos{\left(2 x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Let $u = 2 x$ .
  3. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  4. 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:
    
    $- 2 \sin{\left(2 x \right)}$
  The result is: $- 2 \sin{\left(2 x \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (1+cos2x)/cosx

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Piecewise:

The solution