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Derivative of:
Derivative of 1/2*x^2
Derivative of -(x^2+1)/x
Derivative of sin(x)-cos(x)
Derivative of e^(1/x)
Identical expressions
-(x^ two + one)/x
minus (x squared plus 1) divide by x
minus (x to the power of two plus one) divide by x
-(x2+1)/x
-x2+1/x
-(x²+1)/x
-(x to the power of 2+1)/x
-x^2+1/x
-(x^2+1) divide by x
Similar expressions
(x^2+1)/x
-(x^2-1)/x

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

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

The solution

You have entered [src]

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

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

(-x^2 - 1)/x

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

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. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $- 2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /          2\
  |     1 + x |
6*|-1 + ------|
  |        2  |
  \       x   /
---------------
        2      
       x

$$\frac{6 \left(-1 + \frac{x^{2} + 1}{x^{2}}\right)}{x^{2}}$$

Simplify

The graph

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