Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of (cos(x))^sin(x)
Derivative of xsinx+cosx
Derivative of y=ln(1/x+sqrt(1+x^2)/x^2)
Derivative of sinx+5cosx
Identical expressions
y=ln(one /x+sqrt(one +x^ two)/x^ two)
y equally ln(1 divide by x plus square root of (1 plus x squared ) divide by x squared )
y equally ln(one divide by x plus square root of (one plus x to the power of two) divide by x to the power of two)
y=ln(1/x+√(1+x^2)/x^2)
y=ln(1/x+sqrt(1+x2)/x2)
y=ln1/x+sqrt1+x2/x2
y=ln(1/x+sqrt(1+x²)/x²)
y=ln(1/x+sqrt(1+x to the power of 2)/x to the power of 2)
y=ln1/x+sqrt1+x^2/x^2
y=ln(1 divide by x+sqrt(1+x^2) divide by x^2)
Similar expressions
y=ln(1/x-sqrt(1+x^2)/x^2)
y=ln(1/x+sqrt(1-x^2)/x^2)

The derivative of the function/ y=ln(1/x+sqrt(1+x^2)/x^2)

Derivative of y=ln(1/x+sqrt(1+x^2)/x^2)

The solution

You have entered [src]

   /         ________\
   |        /      2 |
   |  1   \/  1 + x  |
log|1*- + -----------|
   |  x         2    |
   \           x     /

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

  /   /         ________\\
  |   |        /      2 ||
d |   |  1   \/  1 + x  ||
--|log|1*- + -----------||
dx|   |  x         2    ||
  \   \           x     //

$$\frac{d}{d x} \log{\left(\frac{\sqrt{x^{2} + 1}}{x^{2}} + 1 \cdot \frac{1}{x} \right)}$$

Transform

Detail solution

Let $u = \frac{\sqrt{x^{2} + 1}}{x^{2}} + 1 \cdot \frac{1}{x}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\frac{\sqrt{x^{2} + 1}}{x^{2}} + 1 \cdot \frac{1}{x}\right)$ :
1. Differentiate $\frac{\sqrt{x^{2} + 1}}{x^{2}} + 1 \cdot \frac{1}{x}$ term by term:
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 1$ and $g{\left(x \right)} = x$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of the constant $1$ is zero.
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    Now plug in to the quotient rule:
    
    $- \frac{1}{x^{2}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sqrt{x^{2} + 1}$ and $g{\left(x \right)} = x^{2}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = x^{2} + 1$ .
    2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
      1. Differentiate $x^{2} + 1$ term by term:
        
        The derivative of the constant $1$ is zero.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        The result is: $2 x$
      The result of the chain rule is:
      
      $\frac{x}{\sqrt{x^{2} + 1}}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    Now plug in to the quotient rule:
    
    $\frac{\frac{x^{3}}{\sqrt{x^{2} + 1}} - 2 x \sqrt{x^{2} + 1}}{x^{4}}$
  The result is: $- \frac{1}{x^{2}} + \frac{\frac{x^{3}}{\sqrt{x^{2} + 1}} - 2 x \sqrt{x^{2} + 1}}{x^{4}}$
The result of the chain rule is:

$\frac{- \frac{1}{x^{2}} + \frac{\frac{x^{3}}{\sqrt{x^{2} + 1}} - 2 x \sqrt{x^{2} + 1}}{x^{4}}}{\frac{\sqrt{x^{2} + 1}}{x^{2}} + 1 \cdot \frac{1}{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

            ________                 
           /      2                  
  1    2*\/  1 + x           x       
- -- - ------------- + --------------
   2          3              ________
  x          x          2   /      2 
                       x *\/  1 + x  
-------------------------------------
                   ________          
                  /      2           
            1   \/  1 + x            
          1*- + -----------          
            x         2              
                     x

$$\frac{- \frac{1}{x^{2}} + \frac{x}{x^{2} \sqrt{x^{2} + 1}} - \frac{2 \sqrt{x^{2} + 1}}{x^{3}}}{\frac{\sqrt{x^{2} + 1}}{x^{2}} + 1 \cdot \frac{1}{x}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                     3                                                                                                                                                                        
    /                       ________\                                                                                    /                       ________\ /                        ________                 \
    |                      /      2 |                                                                                    |                      /      2 | |                       /      2                  |
    |1        1        2*\/  1 + x  |                                                                                    |1        1        2*\/  1 + x  | |     1        2    6*\/  1 + x           3       |
  2*|- - ----------- + -------------|                                                                                3*x*|- - ----------- + -------------|*|----------- - -- - ------------- + --------------|
    |x      ________          2     |        /                                          ________                 \       |x      ________          2     | |        3/2    3          4              ________|
    |      /      2          x      |        |                                         /      2                  |       |      /      2          x      | |/     2\      x          x          2   /      2 |
    \    \/  1 + x                  /        |  2         x              1         8*\/  1 + x           4       |       \    \/  1 + x                  / \\1 + x /                           x *\/  1 + x  /
- ------------------------------------ + 3*x*|- -- + ----------- + ------------- - ------------- + --------------| - -----------------------------------------------------------------------------------------
                            2                |   4           5/2             3/2          5              ________|                                               ________                                     
           /       ________\                 |  x    /     2\        /     2\            x          3   /      2 |                                              /      2                                      
           |      /      2 |                 \       \1 + x /      x*\1 + x /                      x *\/  1 + x  /                                            \/  1 + x                                       
           |    \/  1 + x  |                                                                                                                              1 + -----------                                     
           |1 + -----------|                                                                                                                                       x                                          
           \         x     /                                                                                                                                                                                  
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                      ________                                                                                                
                                                                                                     /      2                                                                                                 
                                                                                                   \/  1 + x                                                                                                  
                                                                                               1 + -----------                                                                                                
                                                                                                        x

$$\frac{3 x \left(\frac{x}{\left(x^{2} + 1\right)^{\frac{5}{2}}} + \frac{1}{x \left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{4}{x^{3} \sqrt{x^{2} + 1}} - \frac{2}{x^{4}} - \frac{8 \sqrt{x^{2} + 1}}{x^{5}}\right) - \frac{3 x \left(- \frac{1}{\sqrt{x^{2} + 1}} + \frac{1}{x} + \frac{2 \sqrt{x^{2} + 1}}{x^{2}}\right) \left(\frac{1}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{3}{x^{2} \sqrt{x^{2} + 1}} - \frac{2}{x^{3}} - \frac{6 \sqrt{x^{2} + 1}}{x^{4}}\right)}{1 + \frac{\sqrt{x^{2} + 1}}{x}} - \frac{2 \left(- \frac{1}{\sqrt{x^{2} + 1}} + \frac{1}{x} + \frac{2 \sqrt{x^{2} + 1}}{x^{2}}\right)^{3}}{\left(1 + \frac{\sqrt{x^{2} + 1}}{x}\right)^{2}}}{1 + \frac{\sqrt{x^{2} + 1}}{x}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=ln(1/x+sqrt(1+x^2)/x^2)

The graph:

Piecewise:

The solution