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(x^ two)*ln(two +x^ two)
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Similar expressions
(x^2)*ln(2-x^2)

The derivative of the function/ (x^2)*ln(2+x^2)

Derivative of (x^2)*ln(2+x^2)

The solution

You have entered [src]

 2    /     2\
x *log\2 + x /

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

x^2*log(2 + 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 $2 x$
$g{\left(x \right)} = \log{\left(x^{2} + 2 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} + 2$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 2\right)$ :
  1. Differentiate $x^{2} + 2$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. Apply the power rule: $x^{2}$ goes to $2 x$
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{2 x}{x^{2} + 2}$
The result is: $\frac{2 x^{3}}{x^{2} + 2} + 2 x \log{\left(x^{2} + 2 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                      3 
       /     2\    2*x  
2*x*log\2 + x / + ------
                       2
                  2 + x

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

Simplify

The second derivative [src]

  /            /         2 \              \
  |          2 |      2*x  |              |
  |         x *|-1 + ------|              |
  |    2       |          2|              |
  | 4*x        \     2 + x /      /     2\|
2*|------ - ---------------- + log\2 + x /|
  |     2             2                   |
  \2 + x         2 + x                    /

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

Simplify

The third derivative [src]

    /                /         2 \\
    |              2 |      4*x  ||
    |             x *|-3 + ------||
    |        2       |          2||
    |     6*x        \     2 + x /|
4*x*|6 - ------ + ----------------|
    |         2             2     |
    \    2 + x         2 + x      /
-----------------------------------
                    2              
               2 + x

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

Simplify

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Derivative of (x^2)*ln(2+x^2)

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Piecewise:

The solution