Derivative of (x+sqrtx)*lnx

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/      ___\       
\x + \/ x /*log(x)

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

(x + sqrt(x))*log(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (x+sqrtx)*lnx

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The solution