Derivative of y=sqrt(1+tg(x+1/x))

The solution

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    ________________
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  /  1 + tan|x + -| 
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$$\sqrt{\tan{\left(x + \frac{1}{x} \right)} + 1}$$

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

Detail solution

Let $u = \tan{\left(x + \frac{1}{x} \right)} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\tan{\left(x + \frac{1}{x} \right)} + 1\right)$ :
1. Differentiate $\tan{\left(x + \frac{1}{x} \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Rewrite the function to be differentiated:
    
    $\tan{\left(x + \frac{1}{x} \right)} = \frac{\sin{\left(x + \frac{1}{x} \right)}}{\cos{\left(x + \frac{1}{x} \right)}}$
  3. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x + \frac{1}{x} \right)}$ and $g{\left(x \right)} = \cos{\left(x + \frac{1}{x} \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = x + \frac{1}{x}$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \frac{1}{x}\right)$ :
      1. Differentiate $x + \frac{1}{x}$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
        
        The result is: $1 - \frac{1}{x^{2}}$
      The result of the chain rule is:
      
      $\left(1 - \frac{1}{x^{2}}\right) \cos{\left(x + \frac{1}{x} \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = x + \frac{1}{x}$ .
    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 $\frac{d}{d x} \left(x + \frac{1}{x}\right)$ :
      1. Differentiate $x + \frac{1}{x}$ term by term:
        
        Apply the power rule: $x$ goes to $1$
        
        Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
        
        The result is: $1 - \frac{1}{x^{2}}$
      The result of the chain rule is:
      
      $- \left(1 - \frac{1}{x^{2}}\right) \sin{\left(x + \frac{1}{x} \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\left(1 - \frac{1}{x^{2}}\right) \sin^{2}{\left(x + \frac{1}{x} \right)} + \left(1 - \frac{1}{x^{2}}\right) \cos^{2}{\left(x + \frac{1}{x} \right)}}{\cos^{2}{\left(x + \frac{1}{x} \right)}}$
  The result is: $\frac{\left(1 - \frac{1}{x^{2}}\right) \sin^{2}{\left(x + \frac{1}{x} \right)} + \left(1 - \frac{1}{x^{2}}\right) \cos^{2}{\left(x + \frac{1}{x} \right)}}{\cos^{2}{\left(x + \frac{1}{x} \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/       2/    1\\ /    1 \
|1 + tan |x + -||*|1 - --|
\        \    x// |     2|
                  \    x /
--------------------------
        ________________  
       /        /    1\   
  2*  /  1 + tan|x + -|   
    \/          \    x/

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

Simplify

The second derivative [src]

                  /                                    2                  \
                  |                            /    1 \  /       2/    1\\|
                  |                            |1 - --| *|1 + tan |x + -|||
                  |             2              |     2|  \        \    x//|
/       2/    1\\ |1    /    1 \     /    1\   \    x /                   |
|1 + tan |x + -||*|-- + |1 - --| *tan|x + -| - ---------------------------|
\        \    x// | 3   |     2|     \    x/          /       /    1\\    |
                  |x    \    x /                    4*|1 + tan|x + -||    |
                  \                                   \       \    x//    /
---------------------------------------------------------------------------
                                ________________                           
                               /        /    1\                            
                              /  1 + tan|x + -|                            
                            \/          \    x/

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

Simplify

The third derivative [src]

                  /                                                                                                          2         3                                            3                             \
                  |                                                                 /    1 \    /    1\     /       2/    1\\  /    1 \      /       2/    1\\ /    1 \     /    1 \  /       2/    1\\    /    1\|
                  |                                                               6*|1 - --|*tan|x + -|   3*|1 + tan |x + -|| *|1 - --|    3*|1 + tan |x + -||*|1 - --|   3*|1 - --| *|1 + tan |x + -||*tan|x + -||
                  |               3                               3                 |     2|    \    x/     \        \    x//  |     2|      \        \    x// |     2|     |     2|  \        \    x//    \    x/|
/       2/    1\\ |  3    /    1 \  /       2/    1\\     /    1 \     2/    1\     \    x /                                   \    x /                        \    x /     \    x /                              |
|1 + tan |x + -||*|- -- + |1 - --| *|1 + tan |x + -|| + 2*|1 - --| *tan |x + -| + --------------------- + ------------------------------ - ---------------------------- - ----------------------------------------|
\        \    x// |   4   |     2|  \        \    x//     |     2|      \    x/              3                                   2               3 /       /    1\\                    /       /    1\\           |
                  |  x    \    x /                        \    x /                          x                    /       /    1\\             2*x *|1 + tan|x + -||                  2*|1 + tan|x + -||           |
                  |                                                                                            8*|1 + tan|x + -||                  \       \    x//                    \       \    x//           |
                  \                                                                                              \       \    x//                                                                                 /
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                                                                                                    ________________                                                                                               
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                                                                                                  /  1 + tan|x + -|                                                                                                
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$$\frac{\left(\tan^{2}{\left(x + \frac{1}{x} \right)} + 1\right) \left(\left(1 - \frac{1}{x^{2}}\right)^{3} \left(\tan^{2}{\left(x + \frac{1}{x} \right)} + 1\right) + 2 \left(1 - \frac{1}{x^{2}}\right)^{3} \tan^{2}{\left(x + \frac{1}{x} \right)} - \frac{3 \left(1 - \frac{1}{x^{2}}\right)^{3} \left(\tan^{2}{\left(x + \frac{1}{x} \right)} + 1\right) \tan{\left(x + \frac{1}{x} \right)}}{2 \left(\tan{\left(x + \frac{1}{x} \right)} + 1\right)} + \frac{3 \left(1 - \frac{1}{x^{2}}\right)^{3} \left(\tan^{2}{\left(x + \frac{1}{x} \right)} + 1\right)^{2}}{8 \left(\tan{\left(x + \frac{1}{x} \right)} + 1\right)^{2}} + \frac{6 \left(1 - \frac{1}{x^{2}}\right) \tan{\left(x + \frac{1}{x} \right)}}{x^{3}} - \frac{3 \left(1 - \frac{1}{x^{2}}\right) \left(\tan^{2}{\left(x + \frac{1}{x} \right)} + 1\right)}{2 x^{3} \left(\tan{\left(x + \frac{1}{x} \right)} + 1\right)} - \frac{3}{x^{4}}\right)}{\sqrt{\tan{\left(x + \frac{1}{x} \right)} + 1}}$$

Simplify

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Derivative of y=sqrt(1+tg(x+1/x))

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