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Identical expressions
x*(ln^ three (x))
x multiply by (ln cubed (x))
x multiply by (ln to the power of three (x))
x*(ln3(x))
x*ln3x
x*(ln³(x))
x*(ln to the power of 3(x))
x(ln^3(x))
x(ln3(x))
xln3x
xln^3x

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

Derivative of x*(ln^3(x))

The solution

You have entered [src]

     3   
x*log (x)

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

x*log(x)^3

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \log{\left(x \right)}^{3}$ ; 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: $\log{\left(x \right)}^{3} + 3 \log{\left(x \right)}^{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                    2                            \
3*\2 - 6*log(x) + 2*log (x) - 3*(-2 + log(x))*log(x)/
-----------------------------------------------------
                           2                         
                          x

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

Simplify

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Identical expressions

Derivative of x*(ln^3(x))

The graph:

Piecewise:

The solution