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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/

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

d /   2/ 2    \\
--\log \x  - 1//
dx

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

Transform

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 $\left(-1\right) 1$ 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