Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of u
Derivative of e^x/x^2
Derivative of cos(x)/x^2
Derivative of cos(sqrt(x))
Integral of d{x}:
(x²+1)/(x²-1)
Graphing y =:
(x²+1)/(x²-1)
Identical expressions
y=(x²+ one)/(x²- one)
y equally (x² plus 1) divide by (x² minus 1)
y equally (x² plus one) divide by (x² minus one)
y=x²+1/x²-1
y=(x²+1) divide by (x²-1)
Similar expressions
y=(x²+1)/(x²+1)
y=(x²-1)/(x²-1)

The derivative of the function/ y=(x²+1)/(x²-1)

Derivative of y=(x²+1)/(x²-1)

The solution

You have entered [src]

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

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

(x^2 + 1)/(x^2 - 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{2} + 1$ and $g{\left(x \right)} = x^{2} - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

$$\frac{12 x \left(\frac{4 x^{2}}{x^{2} - 1} - 2 - \frac{2 \left(x^{2} + 1\right) \left(\frac{2 x^{2}}{x^{2} - 1} - 1\right)}{x^{2} - 1}\right)}{\left(x^{2} - 1\right)^{2}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=(x²+1)/(x²-1)

The graph:

Piecewise:

The solution