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Derivative of:
Derivative of arctg(1/x)
Derivative of ln(x+3)^3
Derivative of cot(1/x)
Derivative of cos(x)-1
Integral of d{x}:
(x^2+1)/x^2
Graphing y =:
(x^2+1)/x^2
Identical expressions
(x^ two + one)/x^ two
(x squared plus 1) divide by x squared
(x to the power of two plus one) divide by x to the power of two
(x2+1)/x2
x2+1/x2
(x²+1)/x²
(x to the power of 2+1)/x to the power of 2
x^2+1/x^2
(x^2+1) divide by x^2
Similar expressions
(x^2-1)/x^2

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

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

The solution

You have entered [src]

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

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

(x^2 + 1)/x^2

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

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. Apply the power rule: $x^{2}$ goes to $2 x$
Now plug in to the quotient rule:

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

$- \frac{2}{x^{3}}$

The answer is:

$- \frac{2}{x^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second 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 third derivative [src]

   /         2\
   |    1 + x |
24*|1 - ------|
   |       2  |
   \      x   /
---------------
        3      
       x

$$\frac{24 \left(1 - \frac{x^{2} + 1}{x^{2}}\right)}{x^{3}}$$

Simplify

The graph

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