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Limit of the function:
x/sqrt(x^2+2*x)
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Similar expressions
x/sqrt(x^2-2*x)

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

Derivative of x/sqrt(x^2+2*x)

The solution

You have entered [src]

      x      
-------------
   __________
  /  2       
\/  x  + 2*x

$$\frac{x}{\sqrt{x^{2} + 2 x}}$$

d /      x      \
--|-------------|
dx|   __________|
  |  /  2       |
  \\/  x  + 2*x /

$$\frac{d}{d x} \frac{x}{\sqrt{x^{2} + 2 x}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1           x*(1 + x)  
------------- - -------------
   __________             3/2
  /  2          / 2      \   
\/  x  + 2*x    \x  + 2*x/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /             /             2\             \
  |             |    5*(1 + x) |             |
  |     (1 + x)*|3 - ----------|            2|
  |             \    x*(2 + x) /   3*(1 + x) |
3*|-1 + ------------------------ + ----------|
  \              2 + x             x*(2 + x) /
----------------------------------------------
                           3/2                
                (x*(2 + x))

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

Simplify

The graph

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Identical expressions

Similar expressions

Derivative of x/sqrt(x^2+2*x)

The graph:

Piecewise:

The solution