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Identical expressions
(sqrt(x^ three)-sqrt(x))lnx
( square root of (x cubed ) minus square root of (x))lnx
( square root of (x to the power of three) minus square root of (x))lnx
(√(x^3)-√(x))lnx
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Similar expressions
(sqrt(x^3)+sqrt(x))lnx

The derivative of the function/ (sqrt(x^3)-sqrt(x))lnx

Derivative of (sqrt(x^3)-sqrt(x))lnx

The solution

You have entered [src]

/   ____        \       
|  /  3      ___|       
\\/  x   - \/ x /*log(x)

$$\left(- \sqrt{x} + \sqrt{x^{3}}\right) \log{\left(x \right)}$$

(sqrt(x^3) - sqrt(x))*log(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (sqrt(x^3)-sqrt(x))lnx

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The solution