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Identical expressions
y= one / two log(x^2- one)+x
y equally 1 divide by 2 logarithm of (x squared minus 1) plus x
y equally one divide by two logarithm of (x squared minus one) plus x
y=1/2log(x2-1)+x
y=1/2logx2-1+x
y=1/2log(x²-1)+x
y=1/2log(x to the power of 2-1)+x
y=1/2logx^2-1+x
y=1 divide by 2log(x^2-1)+x
Similar expressions
y=1/2log(x^2+1)+x
y=1/2log(x^2-1)-x

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

Derivative of y=1/2log(x^2-1)+x

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Differentiate $x + \frac{\log{\left(x^{2} - 1 \right)}}{2}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $\frac{2 x}{x^{2} - 1}$
  So, the result is: $\frac{x}{x^{2} - 1}$
2. Apply the power rule: $x$ goes to $1$
The result is: $\frac{x}{x^{2} - 1} + 1$
Now simplify:

$\frac{x^{2} + x - 1}{x^{2} - 1}$

The answer is:

$\frac{x^{2} + x - 1}{x^{2} - 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      x   
1 + ------
     2    
    x  - 1

$$\frac{x}{x^{2} - 1} + 1$$

Simplify

The second derivative [src]

         2 
      2*x  
1 - -------
          2
    -1 + x 
-----------
        2  
  -1 + x

$$\frac{- \frac{2 x^{2}}{x^{2} - 1} + 1}{x^{2} - 1}$$

Simplify

The third derivative [src]

    /          2 \
    |       4*x  |
2*x*|-3 + -------|
    |           2|
    \     -1 + x /
------------------
             2    
    /      2\     
    \-1 + x /

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

Simplify

The graph

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Derivative of y=1/2log(x^2-1)+x

The graph:

Piecewise:

The solution