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Integral of d{x}:
(tgx^2+sinx)/cosx^2
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Similar expressions
(tgx^2-sinx)/cosx^2

The derivative of the function/ (tgx^2+sinx)/cosx^2

Derivative of (tgx^2+sinx)/cosx^2

The solution

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

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

(tan(x)^2 + sin(x))/cos(x)^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\sin{\left(x \right)} + \tan^{2}{\left(x \right)}$ term by term:
  1. Let $u = \tan{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
    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 of the chain rule is:
    
    $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  4. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \cos{\left(x \right)}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/         2   \                     /   2            \       
\2 + 2*tan (x)/*tan(x) + cos(x)   2*\tan (x) + sin(x)/*sin(x)
------------------------------- + ---------------------------
               2                               3             
            cos (x)                         cos (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                                                                                                                                         /         2   \                          
                                                                                                                                                                                         |    3*sin (x)| /   2            \       
                                                                                                                       /                         2                          \          8*|2 + ---------|*\tan (x) + sin(x)/*sin(x)
            /         2   \                                                                               2            |            /       2   \         2    /       2   \|            |        2    |                          
            |    3*sin (x)| /  /       2   \                \        3    /       2   \      /       2   \           6*\-sin(x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)//*sin(x)     \     cos (x) /                          
-cos(x) + 6*|1 + ---------|*\2*\1 + tan (x)/*tan(x) + cos(x)/ + 8*tan (x)*\1 + tan (x)/ + 16*\1 + tan (x)/ *tan(x) + --------------------------------------------------------------- + -------------------------------------------
            |        2    |                                                                                                                       cos(x)                                                  cos(x)                  
            \     cos (x) /                                                                                                                                                                                                       
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                2                                                                                                                 
                                                                                                             cos (x)

$$\frac{6 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \cos{\left(x \right)}\right) \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{8 \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \left(\sin{\left(x \right)} + \tan^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} + 16 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + \frac{6 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} - \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$

Simplify

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Derivative of (tgx^2+sinx)/cosx^2

The graph:

Piecewise:

The solution