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Derivative of:
Derivative of cos(x/3)
Derivative of √x
Derivative of xcos(x)
Derivative of sqrt(tgx)/(x^2+1)
Identical expressions
sqrt(tgx)/(x^ two + one)
square root of (tgx) divide by (x squared plus 1)
square root of (tgx) divide by (x to the power of two plus one)
√(tgx)/(x^2+1)
sqrt(tgx)/(x2+1)
sqrttgx/x2+1
sqrt(tgx)/(x²+1)
sqrt(tgx)/(x to the power of 2+1)
sqrttgx/x^2+1
sqrt(tgx) divide by (x^2+1)
Similar expressions
sqrt(tgx)/(x^2-1)

The derivative of the function/ sqrt(tgx)/(x^2+1)

Derivative of sqrt(tgx)/(x^2+1)

The solution

You have entered [src]

  ________
\/ tan(x) 
----------
   2      
  x  + 1

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

sqrt(tan(x))/(x^2 + 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \tan{\left(x \right)}$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} \sqrt{\tan{\left(x \right)}}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           2                        
    1   tan (x)                     
    - + -------             ________
    2      2          2*x*\/ tan(x) 
------------------- - --------------
/ 2    \   ________             2   
\x  + 1/*\/ tan(x)      / 2    \    
                        \x  + 1/

$$- \frac{2 x \sqrt{\tan{\left(x \right)}}}{\left(x^{2} + 1\right)^{2}} + \frac{\frac{\tan^{2}{\left(x \right)}}{2} + \frac{1}{2}}{\left(x^{2} + 1\right) \sqrt{\tan{\left(x \right)}}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

              /                                                2\                                                                                                                   
              |                 /       2   \     /       2   \ |                   /         2 \                   /         2 \                     /                        2   \
/       2   \ |      3/2      4*\1 + tan (x)/   3*\1 + tan (x)/ |          ________ |      2*x  |     /       2   \ |      4*x  |       /       2   \ |      ________   1 + tan (x)|
\1 + tan (x)/*|16*tan   (x) - --------------- + ----------------|   24*x*\/ tan(x) *|-1 + ------|   3*\1 + tan (x)/*|-1 + ------|   3*x*\1 + tan (x)/*|- 4*\/ tan(x)  + -----------|
              |                    ________           5/2       |                   |          2|                   |          2|                     |                     3/2    |
              \                  \/ tan(x)         tan   (x)    /                   \     1 + x /                   \     1 + x /                     \                  tan   (x) /
----------------------------------------------------------------- - ----------------------------- + ----------------------------- + ------------------------------------------------
                                8                                                     2                  /     2\   ________                             /     2\                   
                                                                              /     2\                   \1 + x /*\/ tan(x)                            2*\1 + x /                   
                                                                              \1 + x /                                                                                              
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                            2                                                                                       
                                                                                       1 + x

$$\frac{\frac{3 x \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{2 \left(x^{2} + 1\right)} - \frac{24 x \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right) \sqrt{\tan{\left(x \right)}}}{\left(x^{2} + 1\right)^{2}} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{\frac{5}{2}}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\sqrt{\tan{\left(x \right)}}} + 16 \tan^{\frac{3}{2}}{\left(x \right)}\right)}{8} + \frac{3 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(x^{2} + 1\right) \sqrt{\tan{\left(x \right)}}}}{x^{2} + 1}$$

Simplify

The graph

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Derivative of sqrt(tgx)/(x^2+1)

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Piecewise:

The solution