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

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  ___           
\/ x *log(x) + x

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

sqrt(x)*log(x) + x

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

-log(x) 
--------
    3/2 
 4*x

$$- \frac{\log{\left(x \right)}}{4 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

-2 + 3*log(x)
-------------
       5/2   
    8*x

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

Simplify