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Identical expressions
y=ln^ two ((x^ two)+ one)
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y=ln^2x^2+1
Similar expressions
y=ln^2((x^2)-1)

The derivative of the function/ y=ln^2((x^2)+1)

Derivative of y=ln^2((x^2)+1)

The solution

You have entered [src]

   2/ 2    \
log \x  + 1/

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

log(x^2 + 1)^2

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

    /                        2       2    /     2\\
    |         /     2\    6*x     4*x *log\1 + x /|
8*x*|3 - 3*log\1 + x / - ------ + ----------------|
    |                         2             2     |
    \                    1 + x         1 + x      /
---------------------------------------------------
                             2                     
                     /     2\                      
                     \1 + x /

$$\frac{8 x \left(\frac{4 x^{2} \log{\left(x^{2} + 1 \right)}}{x^{2} + 1} - \frac{6 x^{2}}{x^{2} + 1} - 3 \log{\left(x^{2} + 1 \right)} + 3\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The graph

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Derivative of y=ln^2((x^2)+1)

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The solution