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The derivative of the function/ 2xcosx/cos^2x

Derivative of 2xcosx/cos^2x

The solution

You have entered [src]

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

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

((2*x)*cos(x))/cos(x)^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result is: $- x \sin{\left(x \right)} + \cos{\left(x \right)}$
  So, the result is: $- 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

2*cos(x) - 2*x*sin(x)   4*x*sin(x)
--------------------- + ----------
          2                 2     
       cos (x)           cos (x)

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

Simplify

The second derivative [src]

  /                            /         2   \                                \
  |  2*sin(x) + x*cos(x)       |    3*sin (x)|   4*(-cos(x) + x*sin(x))*sin(x)|
2*|- ------------------- + 2*x*|1 + ---------| - -----------------------------|
  |         cos(x)             |        2    |                 2              |
  \                            \     cos (x) /              cos (x)           /
-------------------------------------------------------------------------------
                                     cos(x)

$$\frac{2 \left(2 x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) - \frac{4 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{x \cos{\left(x \right)} + 2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos{\left(x \right)}}$$

Simplify

The third derivative [src]

  /                         /         2   \                                                             /         2   \       \
  |                         |    3*sin (x)|                        6*(2*sin(x) + x*cos(x))*sin(x)       |    3*sin (x)|       |
2*|-3*cos(x) + x*sin(x) - 6*|1 + ---------|*(-cos(x) + x*sin(x)) - ------------------------------ + 8*x*|2 + ---------|*sin(x)|
  |                         |        2    |                                    cos(x)                   |        2    |       |
  \                         \     cos (x) /                                                             \     cos (x) /       /
-------------------------------------------------------------------------------------------------------------------------------
                                                               2                                                               
                                                            cos (x)

$$\frac{2 \left(8 x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin{\left(x \right)} + x \sin{\left(x \right)} - 6 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) - \frac{6 \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - 3 \cos{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}}$$

Simplify

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Identical expressions

Derivative of 2xcosx/cos^2x

The graph:

Piecewise:

The solution