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Identical expressions
one two x^ two + one /tgx*1/cosx^2
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12x2+1/tgx*1/cosx2
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Similar expressions
12x^2-1/tgx*1/cosx^2

The derivative of the function/ 12x^2+1/tgx*1/cosx^2

Derivative of 12x^2+1/tgx*1/cosx^2

The solution

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

$$12 x^{2} + \frac{1}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$$

12*x^2 + 1/(tan(x)*cos(x)^2)

Detail solution

Differentiate $12 x^{2} + \frac{1}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  So, the result is: $24 x$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 1$ and $g{\left(x \right)} = \cos^{2}{\left(x \right)} \tan{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of the constant $1$ is zero.
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \cos^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    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:
      
      $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
    $g{\left(x \right)} = \tan{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    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:
      
      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
    The result is: $\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} + \cos^{2}{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - \cos^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)} \tan^{2}{\left(x \right)}}$
The result is: $24 x + \frac{- \sin^{2}{\left(x \right)} + 2 \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} - \cos^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)} \tan^{2}{\left(x \right)}}$
Now simplify:

$\frac{48 x \cos^{2}{\left(2 x \right)} - 48 x + 8 \cos{\left(2 x \right)}}{\cos{\left(4 x \right)} - 1}$

The answer is:

$\frac{48 x \cos^{2}{\left(2 x \right)} - 48 x + 8 \cos{\left(2 x \right)}}{\cos{\left(4 x \right)} - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 2                     
         -1 - tan (x)       2*sin(x)   
24*x + --------------- + --------------
          2       2         3          
       cos (x)*tan (x)   cos (x)*tan(x)

$$24 x + \frac{- \tan^{2}{\left(x \right)} - 1}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{\cos^{3}{\left(x \right)} \tan{\left(x \right)}}$$

Simplify

The second derivative [src]

  /                                    2                                                           \
  |                       /       2   \            2               2           /       2   \       |
  |           1           \1 + tan (x)/     1 + tan (x)       3*sin (x)      2*\1 + tan (x)/*sin(x)|
2*|12 + -------------- + --------------- - -------------- + -------------- - ----------------------|
  |        2                2       3         2                4                   3       2       |
  \     cos (x)*tan(x)   cos (x)*tan (x)   cos (x)*tan(x)   cos (x)*tan(x)      cos (x)*tan (x)    /

$$2 \left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\cos^{2}{\left(x \right)} \tan^{3}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)} \tan^{2}{\left(x \right)}} - \frac{\tan^{2}{\left(x \right)} + 1}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)} \tan{\left(x \right)}} + 12 + \frac{1}{\cos^{2}{\left(x \right)} \tan{\left(x \right)}}\right)$$

Simplify

The third derivative [src]

  /                                3                                    2                                                                                                      2       \
  |                   /       2   \      /       2   \     /       2   \                            3             2    /       2   \     /       2   \            /       2   \        |
  |          2      3*\1 + tan (x)/    3*\1 + tan (x)/   5*\1 + tan (x)/       8*sin(x)       12*sin (x)     9*sin (x)*\1 + tan (x)/   6*\1 + tan (x)/*sin(x)   6*\1 + tan (x)/ *sin(x)|
2*|-2 - 2*tan (x) - ---------------- - --------------- + ---------------- + ------------- + -------------- - ----------------------- - ---------------------- + -----------------------|
  |                        4                  2                 2           cos(x)*tan(x)      3                    2       2              cos(x)*tan(x)                       3       |
  \                     tan (x)            tan (x)           tan (x)                        cos (x)*tan(x)       cos (x)*tan (x)                                     cos(x)*tan (x)    /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                           2                                                                                            
                                                                                        cos (x)

$$\frac{2 \left(- \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(x \right)}}{\cos{\left(x \right)} \tan^{3}{\left(x \right)}} + \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} - \frac{9 \left(\tan^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + \frac{12 \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)} \tan{\left(x \right)}} + \frac{8 \sin{\left(x \right)}}{\cos{\left(x \right)} \tan{\left(x \right)}} - 2 \tan^{2}{\left(x \right)} - 2\right)}{\cos^{2}{\left(x \right)}}$$

Simplify

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Derivative of 12x^2+1/tgx*1/cosx^2

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