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

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Derivative of:
Derivative of x^5*sin(x)
Derivative of (x^2-x)/2
Derivative of (x^2-1)/((2*x))
Derivative of x^(-1/3)
Graphing y =:
(x^2-1)/((2*x))
Identical expressions
(x^ two - one)/((two *x))
(x squared minus 1) divide by ((2 multiply by x))
(x to the power of two minus one) divide by ((two multiply by x))
(x2-1)/((2*x))
x2-1/2*x
(x²-1)/((2*x))
(x to the power of 2-1)/((2*x))
(x^2-1)/((2x))
(x2-1)/((2x))
x2-1/2x
x^2-1/2x
(x^2-1) divide by ((2*x))
Similar expressions
(x^2+1)/((2*x))

The derivative of the function/ (x^2-1)/((2*x))

Derivative of (x^2-1)/((2*x))

The solution

You have entered [src]

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

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

  / 2    \
d |x  - 1|
--|------|
dx\ 2*x  /

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

Transform

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 $g{\left(x \right)} = 2 x$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x$ goes to $1$
  So, the result is: $2$
Now plug in to the quotient rule:

$\frac{2 x^{2} + 2}{4 x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of (x^2-1)/((2*x))