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The derivative of the function/ y=xln²x

Derivative of y=xln²x

The solution

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     2   
x*log (x)

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

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

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

2*(1 + log(x))
--------------
      x

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

Simplify

The third derivative [src]

-2*log(x)
---------
     2   
    x

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

Simplify

The graph

Derivative of y=xln²x