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Identical expressions
((sqrt(x))+ three)/(x+ two)
(( square root of (x)) plus 3) divide by (x plus 2)
(( square root of (x)) plus three) divide by (x plus two)
((√(x))+3)/(x+2)
sqrtx+3/x+2
((sqrt(x))+3) divide by (x+2)
Similar expressions
((sqrt(x))+3)/(x-2)
((sqrt(x))-3)/(x+2)

The derivative of the function/ ((sqrt(x))+3)/(x+2)

Derivative of ((sqrt(x))+3)/(x+2)

The solution

You have entered [src]

  ___    
\/ x  + 3
---------
  x + 2

$$\frac{\sqrt{x} + 3}{x + 2}$$

(sqrt(x) + 3)/(x + 2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{x} + 3$ and $g{\left(x \right)} = x + 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\sqrt{x} + 3$ term by term:
  1. The derivative of the constant $3$ is zero.
  2. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  The result is: $\frac{1}{2 \sqrt{x}}$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ((sqrt(x))+3)/(x+2)

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The solution