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y=((sqrt one +tgx+1/x))
y equally (( square root of 1 plus tgx plus 1 divide by x))
y equally (( square root of one plus tgx plus 1 divide by x))
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y=sqrt1+tgx+1/x
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Similar expressions
y=sqrt(1+tg*(x+1/x)).
y=(sqrt1+tg(x+1/x))
y=((sqrt1-tgx+1/x))
y=((sqrt1+tgx-1/x))
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The derivative of the function/ y=((sqrt1+tgx+1/x))

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

The solution

You have entered [src]

  ___            1
\/ 1  + tan(x) + -
                 x

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

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

Detail solution

Differentiate $\left(\tan{\left(x \right)} + \sqrt{1}\right) + \frac{1}{x}$ term by term:
1. Differentiate $\tan{\left(x \right)} + \sqrt{1}$ term by term:
  1. The derivative of the constant $\sqrt{1}$ is zero.
  2. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  3. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ 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: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
2. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{1}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2      1 
1 + tan (x) - --
               2
              x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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

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The solution