Derivative of y=sqrt1+tg(x+(1/x))

The solution

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  ___      /    1\
\/ 1  + tan|x + -|
           \    x/

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

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

Detail solution

Differentiate $\tan{\left(x + \frac{1}{x} \right)} + \sqrt{1}$ term by term:
1. The derivative of the constant $\sqrt{1}$ 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:
      1. Apply the power rule: $x$ goes to $1$
      2. 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:
      1. Apply the power rule: $x$ goes to $1$
      2. 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: $\frac{\left(1 - \frac{1}{x^{2}}\right) \sin^{2}{\left(x + \frac{1}{x} \right)} + \left(1 - \frac{1}{x^{2}}\right) \cos^{2}{\left(x + \frac{1}{x} \right)}}{\cos^{2}{\left(x + \frac{1}{x} \right)}}$
Now simplify:

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

The answer is:

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

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 /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                    /                                                                 /    1 \    /    1\\
                    |                                                               6*|1 - --|*tan|x + -||
                    |               3                               3                 |     2|    \    x/|
  /       2/    1\\ |  3    /    1 \  /       2/    1\\     /    1 \     2/    1\     \    x /           |
2*|1 + tan |x + -||*|- -- + |1 - --| *|1 + tan |x + -|| + 2*|1 - --| *tan |x + -| + ---------------------|
  \        \    x// |   4   |     2|  \        \    x//     |     2|      \    x/              3         |
                    \  x    \    x /                        \    x /                          x          /

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

Simplify

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Derivative of y=sqrt1+tg(x+(1/x))

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