Derivative of cos2x/cosx

The solution

You have entered [src]

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

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

cos(2*x)/cos(x)

Detail solution

Apply the quotient rule, which is:

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

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

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. 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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of cos2x/cosx

The graph:

Piecewise:

The solution