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

You have entered [src]

  ___
\/ x 
-----
x + 2

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

  /  ___\
d |\/ x |
--|-----|
dx\x + 2/

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x + 2$ term by term:
  1. The derivative of the constant $2$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     ___  
       1           \/ x   
--------------- - --------
    ___                  2
2*\/ x *(x + 2)   (x + 2)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                              ___                  \
  |  1            1          2*\/ x           1       |
3*|------ + -------------- - -------- + --------------|
  |   5/2     ___        2          3      3/2        |
  \8*x      \/ x *(2 + x)    (2 + x)    4*x   *(2 + x)/
-------------------------------------------------------
                         2 + x

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution