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Derivative of:
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Identical expressions
y=(ln^ three (x))x
y equally (ln cubed (x))x
y equally (ln to the power of three (x))x
y=(ln3(x))x
y=ln3xx
y=(ln³(x))x
y=(ln to the power of 3(x))x
y=ln^3xx

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

Derivative of y=(ln^3(x))x

The solution

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   3     
log (x)*x

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

d /   3     \
--\log (x)*x/
dx

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

Transform

Detail solution

Apply the product rule:

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

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

The graph

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Identical expressions

Derivative of y=(ln^3(x))x

The graph:

Piecewise:

The solution