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

Derivative of x*sqrt(x+3)

The solution

You have entered [src]

    _______
x*\/ x + 3

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

x*sqrt(x + 3)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  _______        x     
\/ x + 3  + -----------
                _______
            2*\/ x + 3

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

Simplify

The second derivative [src]

        x    
1 - ---------
    4*(3 + x)
-------------
    _______  
  \/ 3 + x

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

Simplify

The third derivative [src]

  /       x  \
3*|-2 + -----|
  \     3 + x/
--------------
          3/2 
 8*(3 + x)

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

Simplify

The graph

Derivative of x*sqrt(x+3)