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Limit of the function:
x^3/log(x)
Identical expressions
x^ three /log(x)
x cubed divide by logarithm of (x)
x to the power of three divide by logarithm of (x)
x3/log(x)
x3/logx
x³/log(x)
x to the power of 3/log(x)
x^3/logx
x^3 divide by log(x)

The derivative of the function/ x^3/log(x)

Derivative of x^3/log(x)

The solution

You have entered [src]

   3  
  x   
------
log(x)

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

x^3/log(x)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{3}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{3 x^{2} \log{\left(x \right)} - x^{2}}{\log{\left(x \right)}^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      2         2 
     x       3*x  
- ------- + ------
     2      log(x)
  log (x)

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

Simplify

The second derivative [src]

  /                   2   \
  |             1 + ------|
  |      6          log(x)|
x*|6 - ------ + ----------|
  \    log(x)     log(x)  /
---------------------------
           log(x)

$$\frac{x \left(\frac{1 + \frac{2}{\log{\left(x \right)}}}{\log{\left(x \right)}} + 6 - \frac{6}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}}$$

Simplify

The third derivative [src]

               /      3         3   \                 
             2*|1 + ------ + -------|     /      2   \
               |    log(x)      2   |   9*|1 + ------|
      18       \             log (x)/     \    log(x)/
6 - ------ - ------------------------ + --------------
    log(x)            log(x)                log(x)    
------------------------------------------------------
                        log(x)

$$\frac{\frac{9 \left(1 + \frac{2}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + 6 - \frac{18}{\log{\left(x \right)}}}{\log{\left(x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of x^3/log(x)

The graph:

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The solution