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

You have entered [src]

    ___________
3*\/ x*(x + 2)

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

3*sqrt(x*(x + 2))

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

3-я производная [src]

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

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

Simplify

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

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