Derivative of y=x^2log^2x

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 2    2   
x *log (x)

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

x^2*log(x)^2

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 $2 x$
$g{\left(x \right)} = \log{\left(x \right)}^{2}$ ; 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: $2 x \log{\left(x \right)}^{2} + 2 x \log{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /       2              \
2*\1 + log (x) + 3*log(x)/

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

Simplify

The third derivative [src]

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

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

Simplify

The graph