Derivative of x^2lnx^3

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 2    3   
x *log (x)

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

d / 2    3   \
--\x *log (x)/
dx

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/         2              \       
\6 + 2*log (x) + 9*log(x)/*log(x)

$$\left(2 \log{\left(x \right)}^{2} + 9 \log{\left(x \right)} + 6\right) \log{\left(x \right)}$$

Simplify

The third derivative [src]

  /                    2                            \
6*\1 - 3*log(x) + 4*log (x) - 3*(-2 + log(x))*log(x)/
-----------------------------------------------------
                          x

$$\frac{6 \left(- 3 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + 4 \log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x}$$

Simplify

The graph

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Identical expressions

Derivative of x^2lnx^3

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The solution