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Derivative of:
Derivative of y=4x
Derivative of 3/x^4
Derivative of 2*x+3
Derivative of 2^x+e^x
Identical expressions
log(y^ two - one)
logarithm of (y squared minus 1)
logarithm of (y to the power of two minus one)
log(y2-1)
logy2-1
log(y²-1)
log(y to the power of 2-1)
logy^2-1
Similar expressions
log(y^2+1)

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

Derivative of log(y^2-1)

The solution

You have entered [src]

   / 2    \
log\y  - 1/

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

log(y^2 - 1)

Detail solution

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

$\frac{2 y}{y^{2} - 1}$
Now simplify:

$\frac{2 y}{y^{2} - 1}$

The answer is:

$\frac{2 y}{y^{2} - 1}$

The graph

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The first derivative [src]

 2*y  
------
 2    
y  - 1

$$\frac{2 y}{y^{2} - 1}$$

Simplify

The second derivative [src]

  /         2 \
  |      2*y  |
2*|1 - -------|
  |          2|
  \    -1 + y /
---------------
          2    
    -1 + y

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log(y^2-1)

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The solution