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Derivative of 3x²-2
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Identical expressions
sqrtln(x)ln^ two (x)ln^3x
square root of ln(x)ln squared (x)ln cubed x
square root of ln(x)ln to the power of two (x)ln cubed x
√ln(x)ln^2(x)ln^3x
sqrtln(x)ln2(x)ln3x
sqrtlnxln2xln3x
sqrtln(x)ln²(x)ln³x
sqrtln(x)ln to the power of 2(x)ln to the power of 3x
sqrtlnxln^2xln^3x

The derivative of the function/ sqrtln(x)ln^2(x)ln^3x

Derivative of sqrtln(x)ln^2(x)ln^3x

The solution

You have entered [src]

  ________    2       3   
\/ log(x) *log (x)*log (x)

$$\sqrt{\log{\left(x \right)}} \log{\left(x \right)}^{2} \log{\left(x \right)}^{3}$$

(sqrt(log(x))*log(x)^2)*log(x)^3

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \sqrt{\log{\left(x \right)}} \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{\log{\left(x \right)}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{1}{2 x \sqrt{\log{\left(x \right)}}}$
  $g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \log{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result of the chain rule is:
    
    $\frac{2 \log{\left(x \right)}}{x}$
  The result is: $\frac{5 \log{\left(x \right)}^{\frac{3}{2}}}{2 x}$
$g{\left(x \right)} = \log{\left(x \right)}^{3}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result of the chain rule is:
  
  $\frac{3 \log{\left(x \right)}^{2}}{x}$
The result is: $\frac{11 \log{\left(x \right)}^{\frac{9}{2}}}{2 x}$

The answer is:

$\frac{11 \log{\left(x \right)}^{\frac{9}{2}}}{2 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      9/2   
11*log   (x)
------------
    2*x

$$\frac{11 \log{\left(x \right)}^{\frac{9}{2}}}{2 x}$$

Simplify

The second derivative [src]

          /                /      1   \       \
          |                |2 + ------|*log(x)|
   7/2    |                \    log(x)/       |
log   (x)*|25 - 5*log(x) - -------------------|
          \                         4         /
-----------------------------------------------
                        2                      
                       x

$$\frac{\left(- \frac{\left(2 + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}}{4} - 5 \log{\left(x \right)} + 25\right) \log{\left(x \right)}^{\frac{7}{2}}}{x^{2}}$$

Simplify

The third derivative [src]

                                                                                                                            3    /     3/2    /       3        6   \        ________                        ________ /      1   \   24*(-1 + log(x))\
                                                                         5/2    /                 /      1   \       \   log (x)*|- log   (x)*|8 + ------- + ------| - 16*\/ log(x) *(-3 + 2*log(x)) + 12*\/ log(x) *|2 + ------| + ----------------|
                                             5/2                    9*log   (x)*|-16 + 8*log(x) + |2 + ------|*log(x)|           |            |       2      log(x)|                                                 \    log(x)/        ________   |
     5/2    /       2              \   45*log   (x)*(-2 + log(x))               \                 \    log(x)/       /           \            \    log (x)         /                                                                   \/ log(x)    /
6*log   (x)*\1 + log (x) - 3*log(x)/ - -------------------------- - -------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------
                                                   2                                        4                                                                                         8                                                              
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                           3                                                                                                                         
                                                                                                                          x

$$\frac{- \frac{45 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)}^{\frac{5}{2}}}{2} - \frac{9 \left(\left(2 + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)} + 8 \log{\left(x \right)} - 16\right) \log{\left(x \right)}^{\frac{5}{2}}}{4} + 6 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right) \log{\left(x \right)}^{\frac{5}{2}} - \frac{\left(12 \left(2 + \frac{1}{\log{\left(x \right)}}\right) \sqrt{\log{\left(x \right)}} + \frac{24 \left(\log{\left(x \right)} - 1\right)}{\sqrt{\log{\left(x \right)}}} - 16 \left(2 \log{\left(x \right)} - 3\right) \sqrt{\log{\left(x \right)}} - \left(8 + \frac{6}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \log{\left(x \right)}^{\frac{3}{2}}\right) \log{\left(x \right)}^{3}}{8}}{x^{3}}$$

Simplify

Other calculators

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Identical expressions

Derivative of sqrtln(x)ln^2(x)ln^3x

The graph:

Piecewise:

The solution