Mister Exam

Lang:

The derivative of the function/ (x+5)*sqrt(x)

Derivative of (x+5)*sqrt(x)

The solution

You have entered [src]

          ___
(x + 5)*\/ x

$$\sqrt{x} \left(x + 5\right)$$

(x + 5)*sqrt(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x + 5$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x + 5$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $5$ is zero.
  The result is: $1$
$g{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result is: $\sqrt{x} + \frac{x + 5}{2 \sqrt{x}}$
Now simplify:

$\frac{3 x + 5}{2 \sqrt{x}}$

The answer is:

$\frac{3 x + 5}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  ___    x + 5 
\/ x  + -------
            ___
        2*\/ x

$$\sqrt{x} + \frac{x + 5}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

    5 + x
1 - -----
     4*x 
---------
    ___  
  \/ x

$$\frac{1 - \frac{x + 5}{4 x}}{\sqrt{x}}$$

Simplify

The third derivative [src]

  /     5 + x\
3*|-2 + -----|
  \       x  /
--------------
       3/2    
    8*x

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

Simplify

The graph

Derivative of (x+5)*sqrt(x)